{"author":{"avatar":"https://nguyencungthanh.github.io/animation_avatar%282%29.PNG","name":"Nguyen Cung Thanh","url":"https://nguyencungthanh.github.io/"},"feed_url":"https://nguyencungthanh.github.io/tags/functional-analysis/feed.json","home_page_url":"https://nguyencungthanh.github.io/tags/functional-analysis/","items":[{"content_html":"\n  \u003cdiv class='e-content'\u003e\n    \u003cp\u003eThe Fourier–Walsh expansion represents functions on $\\Omega_n=\\lbrace-1,1\\rbrace^n$ in an orthonormal basis of coordinate products. The construction proceeds from a measure space to its Hilbert space $L^2$, then uses orthogonality to derive the coefficients and approximation formulas. On the Boolean cube, the integrals defining the inner product become finite averages.\u003c/p\u003e\n\u003cp\u003eThe construction uses real scalars. Background on measurable functions and integration is given in \u003ca href=\"/2026/05/02/abstract-measure-theory/\"  \u003eAbstract measure theory\u003c/a\u003e; the geometry of inner products is discussed in \u003ca href=\"/2026/04/27/hilbert-space/\"  \u003eHilbert Space\u003c/a\u003e.\u003c/p\u003e\n\u003ch2 id=\"1-from-a-set-to-a-space-of-functions\"\u003e1. From a Set to a Space of Functions\u003c/h2\u003e\n\u003ch3 id=\"11-pointwise-vector-operations\"\u003e1.1 Pointwise vector operations\u003c/h3\u003e\n\u003cp\u003eLet $\\Omega$ be an arbitrary set. The functions $f:\\Omega\\to\\mathbb R$ form a vector space $\\mathcal{V}(\\Omega)$ under pointwise operations: $(f+g)(x)=f(x)+g(x)$ and $(\\alpha f)(x)=\\alpha f(x)$ for $x\\in\\Omega$ and $\\alpha\\in\\mathbb R$. Both operations use the vector space structure of the codomain $\\mathbb R$.\u003c/p\u003e\n\u003cp\u003eIf $\\Omega$ has $m$ elements, ordering them identifies a function with its vector of values in $\\mathbb R^m$, so $\\dim\\mathcal{V}(\\Omega)=m$. For a general domain, an inner product can be constructed by integrating products of functions, after imposing measurability and integrability conditions.\u003c/p\u003e\n\u003ch3 id=\"12-measures-and-integration\"\u003e1.2 Measures and integration\u003c/h3\u003e\n\u003cp\u003eA \u003cstrong\u003emeasure space\u003c/strong\u003e $(\\Omega,\\mathcal A,\\mu)$ consists of a set $\\Omega$, a $\\sigma$-algebra $\\mathcal A$ of its subsets, and a measure $\\mu:\\mathcal A\\to[0,\\infty]$. The $\\sigma$-algebra contains $\\Omega$ and is closed under complements and countable unions. The measure is countably additive on disjoint measurable sets and satisfies $\\mu(\\varnothing)=0$.\u003c/p\u003e\n\u003cp\u003eThis structure defines the integral $\\int_\\Omega f(x)\\,\\mathrm{d}\\mu(x)$ for measurable functions whenever the integral exists. If $\\mu(\\Omega)=1$, then $\\mu$ is a probability measure, and integration gives the expectation of a function under that probability distribution.\u003c/p\u003e\n\u003ch2 id=\"2-constructing-the-hilbert-space-l2\"\u003e2. Constructing the Hilbert Space $L^2$\u003c/h2\u003e\n\u003ch3 id=\"21-square-integrability-and-equivalence-classes\"\u003e2.1 Square integrability and equivalence classes\u003c/h3\u003e\n\u003cp\u003eFix a measure space $(\\Omega,\\mathcal A,\\mu)$. The measurable functions $f:\\Omega\\to\\mathbb R$ satisfying $\\int_\\Omega|f(x)|^2\\,\\mathrm{d}\\mu(x)\u0026lt;\\infty$ form a vector space, denoted by $\\mathcal V_2(\\Omega,\\mu)$.\u003c/p\u003e\n\u003cp\u003eTwo such functions are identified when they agree outside a measurable set of measure zero. This is equality \u003cstrong\u003ealmost everywhere\u003c/strong\u003e with respect to $\\mu$. The space $L^2(\\Omega,\\mu)$ consists of the resulting equivalence classes; a function symbol denotes a representative of its class.\u003c/p\u003e\n\u003cp\u003eFor $f,g\\in L^2(\\Omega,\\mu)$, define\u003c/p\u003e\n\u003cp\u003e$$\\langle f,g\\rangle=\\int_\\Omega f(x)g(x)\\,\\mathrm{d}\\mu(x).$$\u003c/p\u003e\n\u003cp\u003eThe integral form of Cauchy–Schwarz ensures that the product is integrable. The integral defines a symmetric bilinear form on equivalence classes, and $\\langle f,f\\rangle=0$ holds if and only if $f=0$ almost everywhere. The form is therefore positive definite. Its induced norm is\u003c/p\u003e\n\u003cp\u003e$$\\lVert f\\rVert_{L^2(\\Omega,\\mu)}=\\left(\\int_\\Omega|f(x)|^2\\,\\mathrm{d}\\mu(x)\\right)^{1/2}.$$\u003c/p\u003e\n\u003cp\u003eOn the original functions, this expression defines a seminorm whose kernel consists of functions that vanish almost everywhere. Over complex scalars, the corresponding inner product uses $f(x)\\overline{g(x)}$ in the integral.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTheorem 2.1.\u003c/strong\u003e For every measure space $(\\Omega,\\mathcal A,\\mu)$, the space $L^2(\\Omega,\\mu)$ is complete in the norm defined above. Consequently, $\\mathcal{H}=L^2(\\Omega,\\mu)$ is a Hilbert space.\u003c/p\u003e\n\u003cp\u003eCompleteness is the $p=2$ case of the Riesz–Fischer theorem; a proof is given in \u003ca href=\"https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch7.pdf\"  rel='noopener noreferrer'\u003eHunter\u0026rsquo;s notes, Theorem 7.10\u003c/a\u003e.\u003c/p\u003e\n\u003ch3 id=\"22-counting-and-lebesgue-measures\"\u003e2.2 Counting and Lebesgue measures\u003c/h3\u003e\n\u003cp\u003eCounting measure on $\\lbrace1,\\ldots,m\\rbrace$ identifies $L^2$ with $\\mathbb R^m$. The inner product is $\\langle f,g\\rangle=\\sum_{i=1}^m f(i)g(i)$, so the identification preserves the Euclidean geometry.\u003c/p\u003e\n\u003cp\u003eCounting measure on $\\mathbb N$ gives the sequence space $\\ell^2$. A sequence $a=(a_1,a_2,\\ldots)$ belongs to this space when $\\sum_{j=1}^\\infty|a_j|^2\u0026lt;\\infty$, and its inner product with $b\\in\\ell^2$ is $\\langle a,b\\rangle=\\sum_{j=1}^\\infty a_jb_j$.\u003c/p\u003e\n\u003cp\u003eOn $[0,1]$ with Lebesgue measure $\\lambda$, the same construction gives $L^2([0,1],\\lambda)$ with inner product $\\langle f,g\\rangle=\\int_{[0,1]}f(x)g(x)\\,\\mathrm{d}\\lambda(x)$. Here elements are classes of measurable functions whose squared absolute values have finite integrals, with equality understood almost everywhere.\u003c/p\u003e\n\u003ch3 id=\"23-the-role-of-p2\"\u003e2.3 The role of $p=2$\u003c/h3\u003e\n\u003cp\u003eFor $1\\le p\u0026lt;\\infty$, the space $L^p(\\Omega,\\mu)$ consists of equivalence classes of measurable functions satisfying $\\int_\\Omega|f(x)|^p\\,\\mathrm{d}\\mu(x)\u0026lt;\\infty$. Its norm is\u003c/p\u003e\n\u003cp\u003e$$\\lVert f\\rVert_{L^p(\\Omega,\\mu)}=\\left(\\int_\\Omega|f(x)|^p\\,\\mathrm{d}\\mu(x)\\right)^{1/p}.$$\u003c/p\u003e\n\u003cp\u003eAt $p=\\infty$, the space $L^\\infty(\\Omega,\\mu)$ consists of classes of essentially bounded measurable functions, and its norm is the essential supremum of $|f|$ on $\\Omega$ with respect to $\\mu$.\u003c/p\u003e\n\u003cp\u003eThe Riesz–Fischer theorem gives completeness for every $1\\le p\\le\\infty$, so these spaces are \u003ca href=\"/2025/10/08/banach-space/\"  \u003eBanach spaces\u003c/a\u003e. At $p=2$, the identity $\\lVert f\\rVert_{L^2(\\Omega,\\mu)}^2=\\langle f,f\\rangle$ supplies the geometry of orthogonality and projection used in the Fourier construction.\u003c/p\u003e\n\u003ch2 id=\"3-orthonormal-coordinates-and-orthogonal-projection\"\u003e3. Orthonormal Coordinates and Orthogonal Projection\u003c/h2\u003e\n\u003cp\u003eLet $\\mathcal{H}$ be a Hilbert space of finite dimension $m$, with inner product $\\langle\\cdot,\\cdot\\rangle$. Its induced norm is $\\lVert v\\rVert_{\\mathcal{H}}=\\sqrt{\\langle v,v\\rangle}$.\u003c/p\u003e\n\u003cp\u003eA family $e_1,\\ldots,e_k$ is \u003cstrong\u003eorthonormal\u003c/strong\u003e if $\\langle e_i,e_j\\rangle=\\delta_{ij}$, where $\\delta_{ij}$ equals $1$ for $i=j$ and $0$ otherwise. Such a family is linearly independent: if $\\sum_i a_ie_i=0$, taking the inner product with $e_j$ gives $a_j=0$. An orthonormal family of $m$ vectors therefore forms a basis of $\\mathcal{H}$.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eProposition 3.1.\u003c/strong\u003e If $e_1,\\ldots,e_m$ form an orthonormal basis of $\\mathcal{H}$, then every $v\\in\\mathcal{H}$ has the unique expansion $v=\\sum_{j=1}^{m}\\langle v,e_j\\rangle e_j$.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e The basis property gives an expansion $v=\\sum_j c_je_j$. Taking the inner product with $e_k$ yields $\\langle v,e_k\\rangle=c_k$ by orthonormality. Thus every coefficient is uniquely determined.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eProposition 3.2.\u003c/strong\u003e Let $\\mathcal{V}=\\operatorname{span}\\lbrace e_1,\\ldots,e_k\\rbrace$, where $e_1,\\ldots,e_k$ are orthonormal. For each $v\\in\\mathcal{H}$, the unique element of $\\mathcal{V}$ minimizing $\\lVert v-u\\rVert_{\\mathcal{H}}$ is $P_{\\mathcal{V}}v=\\sum_{j=1}^{k}\\langle v,e_j\\rangle e_j$.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e Set $p=P_{\\mathcal{V}}v$. For each $j\\le k$, orthonormality gives $\\langle v-p,e_j\\rangle=0$. Hence $v-p$ is orthogonal to $\\mathcal{V}$. For any $u\\in\\mathcal{V}$, the decomposition $v-u=(v-p)+(p-u)$ is orthogonal, and Pythagoras gives\u003c/p\u003e\n\u003cp\u003e$$\\lVert v-u\\rVert_{\\mathcal{H}}^2=\\lVert v-p\\rVert_{\\mathcal{H}}^2+\\lVert p-u\\rVert_{\\mathcal{H}}^2.$$\u003c/p\u003e\n\u003cp\u003eThe minimum is attained precisely when $u=p$.\u003c/p\u003e\n\u003ch2 id=\"4-fourierwalsh-analysis-on-the-boolean-cube\"\u003e4. Fourier–Walsh Analysis on the Boolean Cube\u003c/h2\u003e\n\u003cp\u003eSet $\\Omega_n=\\lbrace-1,1\\rbrace^n$, take the full $\\sigma$-algebra $\\mathcal A=2^{\\Omega_n}$, and assign each point probability $\\mu(\\lbrace x\\rbrace)=2^{-n}$. Every function on the cube is measurable and square integrable. Since every point has positive probability, the only set of measure zero is the empty set. Equality almost everywhere is therefore pointwise equality, so $L^2(\\Omega_n,\\mu)$ is identified with the entire vector space $\\mathcal{V}(\\Omega_n)$.\u003c/p\u003e\n\u003cp\u003eIn this setting, the integral of any function is a finite average:\u003c/p\u003e\n\u003cp\u003e$$\\int_{\\Omega_n}f(x)\\,\\mathrm{d}\\mu(x)=2^{-n}\\sum_{x\\in\\Omega_n}f(x).$$\u003c/p\u003e\n\u003cp\u003eFor the rest of the article, write $\\mathcal{H}=L^2(\\Omega_n,\\mu)$. If $X=(X_1,\\ldots,X_n)$ is uniformly distributed on $\\Omega_n$, the inner product becomes\u003c/p\u003e\n\u003cp\u003e$$\\langle f,g\\rangle=\\mathbb E[f(X)g(X)]=2^{-n}\\sum_{x\\in\\Omega_n}f(x)g(x).$$\u003c/p\u003e\n\u003cp\u003eThe notation $\\lVert f\\rVert_2$ denotes the $L^2(\\Omega_n,\\mu)$ norm, so $\\lVert f\\rVert_2=(\\mathbb E[f(X)^2])^{1/2}$. The space has dimension $2^n$. The coordinates of $X$ are independent and satisfy $\\mathbb E[X_i]=0$ and $\\mathbb E[X_i^2]=1$.\u003c/p\u003e\n\u003cp\u003eIt remains to construct $2^n$ orthonormal functions.\u003c/p\u003e\n\u003ch3 id=\"41-constructing-the-walsh-basis\"\u003e4.1 Constructing the Walsh Basis\u003c/h3\u003e\n\u003cp\u003eFor a single coordinate with the uniform distribution on $\\lbrace -1,1\\rbrace$, the functions $1$ and $x$ are orthonormal: their squared norms equal $1$, and their inner product is $\\mathbb E[X]=0$. A product construction extends these functions to $n$ coordinates. At coordinate $i$, choose either $1$ or $x_i$, and multiply the choices. A subset $S\\subseteq[n]$, where $[n]=\\lbrace1,\\ldots,n\\rbrace$, records the coordinates contributing $x_i$.\u003c/p\u003e\n\u003cp\u003eFor each $S\\subseteq[n]$, define the \u003cstrong\u003eWalsh function\u003c/strong\u003e $\\chi_S(x)=\\prod_{i\\in S}x_i$, with the convention $\\chi_\\varnothing(x)=1$. These functions are also called \u003cstrong\u003eparity functions\u003c/strong\u003e: $\\chi_S(x)$ equals $1$ when an even number of coordinates indexed by $S$ are negative, and $-1$ when that number is odd.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTheorem 4.1.\u003c/strong\u003e The family $\\lbrace\\chi_S:S\\subseteq[n]\\rbrace$ is an orthonormal basis of $\\mathcal{H}$.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e Since $x_i^2=1$, the product of two Walsh functions satisfies $\\chi_S(x)\\chi_T(x)=\\chi_{S\\triangle T}(x)$, where $S\\triangle T$ is the symmetric difference of the two sets. If $S=T$, this product is identically $1$, so $\\langle\\chi_S,\\chi_T\\rangle=1$. If $S\\ne T$, independence and the fact that each coordinate has mean zero imply\u003c/p\u003e\n\u003cp\u003e$$\\langle\\chi_S,\\chi_T\\rangle=\\mathbb E[\\chi_{S\\triangle T}(X)]=\\prod_{i\\in S\\triangle T}\\mathbb E[X_i]=0.$$\u003c/p\u003e\n\u003cp\u003eThus $\\langle\\chi_S,\\chi_T\\rangle=\\delta_{ST}$. There are $2^n$ subsets of $[n]$, giving $2^n$ orthonormal functions in a space of dimension $2^n$. The family therefore forms a basis.\u003c/p\u003e\n\u003ch3 id=\"42-the-fourierwalsh-expansion\"\u003e4.2 The Fourier–Walsh Expansion\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003eTheorem 4.2.\u003c/strong\u003e Every function $f:\\Omega_n\\to\\mathbb R$ has the unique representation\u003c/p\u003e\n\u003cp\u003e$$f(x)=\\sum_{S\\subseteq[n]}\\widehat f(S)\\chi_S(x),$$\u003c/p\u003e\n\u003cp\u003ewhere $\\widehat f(S)=\\langle f,\\chi_S\\rangle=\\mathbb E[f(X)\\chi_S(X)]$.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e Apply Proposition 3.1 to the basis in Theorem 4.1.\u003c/p\u003e\n\u003cp\u003eThe sum is finite and holds at every point of $\\Omega_n$. Each coordinate occurs at most once in a Walsh function, so the expansion is a \u003cstrong\u003emultilinear polynomial\u003c/strong\u003e. It is the unique multilinear polynomial agreeing with $f$ on the cube. The relation $x_i^2=1$ allows distinct polynomials to represent the same function on this domain.\u003c/p\u003e\n\u003cp\u003eThe constant coefficient is $\\widehat f(\\varnothing)=\\mathbb E[f(X)]$. For a singleton set, $\\widehat f(\\lbrace i\\rbrace)=\\mathbb E[f(X)X_i]$ measures alignment with coordinate $i$. Larger sets index products involving several coordinates. The degree of $\\chi_S$ is $|S|$.\u003c/p\u003e\n\u003cp\u003eFor a Boolean function $f:\\Omega_n\\to\\lbrace-1,1\\rbrace$, the product $f(X)\\chi_S(X)$ equals $1$ on agreement and $-1$ on disagreement. Consequently,\u003c/p\u003e\n\u003cp\u003e$$\\widehat f(S)=2\\mathbb P\\bigl(f(X)=\\chi_S(X)\\bigr)-1.$$\u003c/p\u003e\n\u003ch3 id=\"43-parsevals-identity-mean-and-variance\"\u003e4.3 Parseval\u0026rsquo;s Identity, Mean and Variance\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003eTheorem 4.3.\u003c/strong\u003e For every $f\\in\\mathcal{H}$,\u003c/p\u003e\n\u003cp\u003e$$\\lVert f\\rVert_2^2=\\mathbb E[f(X)^2]=\\sum_{S\\subseteq[n]}\\widehat f(S)^2.$$\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e Substituting the Fourier expansion into the inner product and using orthonormality yields\u003c/p\u003e\n\u003cp\u003e$$\\langle f,f\\rangle=\\sum_{S,T\\subseteq[n]}\\widehat f(S)\\widehat f(T)\\langle\\chi_S,\\chi_T\\rangle=\\sum_{S\\subseteq[n]}\\widehat f(S)^2.$$\u003c/p\u003e\n\u003cp\u003eThe same argument gives $\\langle f,g\\rangle=\\sum_{S\\subseteq[n]}\\widehat f(S)\\widehat g(S)$ for $f,g\\in\\mathcal{H}$. The Fourier transform preserves this inner product when the coefficient space carries the Euclidean inner product.\u003c/p\u003e\n\u003cp\u003eSince $\\widehat f(\\varnothing)=\\mathbb E[f(X)]$, subtracting the mean removes the constant Walsh term. Applying Parseval\u0026rsquo;s identity to the remaining expansion gives\u003c/p\u003e\n\u003cp\u003e$$\\operatorname{Var}(f(X))=\\sum_{\\varnothing\\ne S\\subseteq[n]}\\widehat f(S)^2.$$\u003c/p\u003e\n\u003cp\u003eFor a Boolean function, $f(X)^2=1$, and hence $\\sum_{S\\subseteq[n]}\\widehat f(S)^2=1$.\u003c/p\u003e\n\u003ch3 id=\"44-approximation-by-polynomials-of-bounded-degree\"\u003e4.4 Approximation by Polynomials of Bounded Degree\u003c/h3\u003e\n\u003cp\u003eFor an integer $0\\le d\\le n$, let $\\mathcal V_d$ be the subspace of $\\mathcal{H}$ spanned by the functions $\\chi_S$ with $|S|\\le d$. Equivalently, $\\mathcal V_d$ consists of functions represented by multilinear polynomials of degree at most $d$. Define the truncated expansion by $f_{\\le d}=\\sum_{|S|\\le d}\\widehat f(S)\\chi_S$, where the sum runs over subsets of $[n]$.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eProposition 4.4.\u003c/strong\u003e The function $f_{\\le d}$ is the unique minimizer of $\\lVert f-g\\rVert_2^2=\\mathbb E[(f(X)-g(X))^2]$ over $g\\in\\mathcal V_d$, and its squared approximation error is\u003c/p\u003e\n\u003cp\u003e$$\\lVert f-f_{\\le d}\\rVert_2^2=\\sum_{|S|\u0026gt;d}\\widehat f(S)^2.$$\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e Proposition 3.2 identifies $f_{\\le d}$ as the orthogonal projection onto $\\mathcal V_d$. The residual contains precisely the omitted Walsh terms, so Parseval\u0026rsquo;s identity gives the error formula.\u003c/p\u003e\n\u003cp\u003eThe minimization uses the uniform $L^2$ norm over $\\mathcal V_d$, with $\\mathbb R$ as the codomain of the approximating function.\u003c/p\u003e\n\u003ch2 id=\"references\"\u003eReferences\u003c/h2\u003e\n\u003cp\u003e[1] O\u0026rsquo;Donnell, R. (2014). \u003ca href=\"https://doi.org/10.1017/CBO9781139814782\"  rel='noopener noreferrer'\u003e\u003cem\u003eAnalysis of Boolean Functions\u003c/em\u003e\u003c/a\u003e. Cambridge University Press. Chapter 1. A \u003ca href=\"https://www.cs.cmu.edu/~odonnell/papers/Analysis-of-Boolean-Functions-by-Ryan-ODonnell.pdf\"  rel='noopener noreferrer'\u003erevised text maintained by the author\u003c/a\u003e is freely available.\u003c/p\u003e\n\u003cp\u003e[2] Hunter, J. K. (2011). \u003ca href=\"https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch7.pdf\"  rel='noopener noreferrer'\u003e\u003cem\u003eMeasure Theory\u003c/em\u003e, Chapter 7: $L^p$ spaces\u003c/a\u003e. University of California, Davis.\u003c/p\u003e\n  \u003c/div\u003e\n\u003cp\u003e\n  \u003ca href=\"mailto:nguyencungthanh11@gmail.com?subject=Re: A%20note%20on%20Fourier%e2%80%93Walsh%20Analysis%20on%20the%20Boolean%20Cube\" rel='noopener noreferrer'\u003e💬 Reply via e-mail\u003c/a\u003e\n\u003c/p\u003e\n","date_modified":"2026-10-29T00:00:00+00:00","date_published":"2026-10-29T00:00:00+00:00","id":"https://nguyencungthanh.github.io/2026/10/29/fourier-walsh-analysis-on-the-boolean-cube/","summary":"The Fourier–Walsh expansion represents functions on $\\Omega_n=\\lbrace-1,1\\rbrace^n$ in an orthonormal basis of coordinate products. The construction proceeds from a measure space to its Hilbert space $L^2$, then uses orthogonality to derive the coefficients and approximation formulas. On the Boolean cube, the integrals defining the inner product become finite averages.\n","tags":["Fourier Analysis","Boolean Functions","Functional Analysis"],"title":"A note on Fourier–Walsh Analysis on the Boolean Cube","url":"https://nguyencungthanh.github.io/2026/10/29/fourier-walsh-analysis-on-the-boolean-cube/"},{"content_html":"\n  \u003cdiv class='e-content'\u003e\n    \u003ccenter\u003e\u003cb\u003eAbstract\u003c/b\u003e\u003c/center\u003e\n\u003cp\u003e$\\quad$ We study representations of functions of inner products via embeddings into Hilbert spaces. While polynomials with nonnegative coefficients admit a single feature map realization, this approach fails when negative coefficients are present. To address this limitation, we introduce a two-map framework. Specifically, for any analytic function $f(x) = \\sum_{k=0}^{\\infty} a_k x^k$ with global convergence, we construct a Hilbert space $\\mathcal{H}$ and transformations $\\Phi, \\Psi: \\mathbb{R}^n \\to \\mathcal{H}$ such that\n$$\\langle \\Phi(u), \\Psi(v) \\rangle = f(\\langle u, v \\rangle)\n\\quad \\text{for all } u, v \\in \\mathbb{R}^n.$$\u003c/p\u003e\n\u003ch2 id=\"1-real-analytic-function\"\u003e1. Real analytic function\u003c/h2\u003e\n\u003cp\u003eWe start with the following problem:\u003c/p\u003e\n\u003cp\u003eLet $P(x)$ be a polynomial with nonnegative coefficients, then there exists a Hilbert space $\\mathcal{H}$ and a transformation $\\Phi : \\mathbb{R}^n \\to \\mathcal{H}$ such that:\n$$\\langle \\Phi(u), \\Phi(v) \\rangle = P(\\langle u, v \\rangle) \\quad \\text{for all } u, v \\in \\mathbb{R^n}.$$\nBut our task can be impossible if $P$ has a negative coefficient since $\\langle \\Phi(u), \\Phi(v) \\rangle$ is always nonnegative. Therefore, a natural question is what if we consider two transformations $\\Phi, \\Psi: \\mathbb{R}^n \\to \\mathcal{H}?$ This leads us to a more general problem about analytic function as follow.\u003c/p\u003e\n\u003cp\u003eConsider a function\n$$\nf(x) = \\sum_{k=0}^{\\infty} a_k x^k\n$$\nwhere the series absolutely converges for all $x \\in \\mathbb{R}$. There exist a Hilbert space $\\mathcal{H}$ and transformations $\\Phi, \\Psi : \\mathbb{R}^n \\to \\mathcal{H}$ such that\n$$\n\\langle \\Phi(u), \\Psi(v) \\rangle = f(\\langle u, v \\rangle)\n\\quad \\text{for all } u, v \\in \\mathbb{R}^n.\n$$\nAlso, for any unit vector $u$, we have\n$$ \\|\\Phi(u)\\|^2_\\mathcal{H} = \\|\\Psi(u)\\|^2_\\mathcal{H} =\\sum _{k=0}^{\\infty} |a_k| .$$\n\u003cem\u003eProof.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eFor each $k \\ge 0$, let $V_k = \\mathbb{R}^{n^k}$ and define\n$$\nH = \\bigoplus_{k=0}^{\\infty} V_k =\n\\left\\{ (w_0, w_1, \\dots) : w_k \\in V_k,\\ \\sum_{k=0}^{\\infty} \\|w_k\\|^2 \u0026lt; \\infty \\right\\}\n$$\u003c/p\u003e\n\u003cp\u003ewith inner product\n$$\n\\langle w, z \\rangle_\\mathcal{H} = \\sum_{k=0}^{\\infty} \\langle w_k, z_k \\rangle_{V_k}.\n$$\u003c/p\u003e\n\u003cp\u003eConsider $s_k = \\operatorname{sign}(a_k)$ and write $a_k = s_k |a_k|$. For $u \\in \\mathbb{R}^n$, define\n$$\n\\Phi(u) = (\\sqrt{|a_0|} u^{\\otimes 0}, \\sqrt{|a_1|} u^{\\otimes 1}, \\dots),\n$$\u003c/p\u003e\n\u003cp\u003e$$\n\\Psi(v) = (s_0 \\sqrt{|a_0|} v^{\\otimes 0}, s_1 \\sqrt{|a_1|} v^{\\otimes 1}, \\dots).\n$$\u003c/p\u003e\n\u003cp\u003eThen we prove that $\\Phi(u), \\Psi(v) \\in \\mathcal{H}$. Indeed, we have:\n$$\n\\left\\|\\sqrt{|a_k|} u^{\\otimes k}\\right\\|^2 = |a_k| \\left\\|u^{\\otimes k}\\right\\|^2 = |a_k| \\|u\\|^{2k}.\n$$\u003c/p\u003e\n\u003cp\u003eThus,\n$$ \\sum_{k=0}^{\\infty}\\left\\|\\sqrt{|a_k|} u^{\\otimes k}\\right\\|^2 = \\sum_{k=0}^{\\infty} |a_k| \\|u\\|^{2k}\n$$\nconverges, so $\\Phi(u) \\in \\mathcal{H}$. Similarly, $\\Psi(v) \\in \\mathcal{H}$.\u003c/p\u003e\n\u003cp\u003eOn the other hand,\n$$\n\\begin{aligned}\n\\langle \\Phi(u), \\Psi(v) \\rangle_\\mathcal{H} \u0026amp;=\n\\sum_{k=0}^{\\infty} \\left\\langle \\sqrt{|a_k|} u^{\\otimes k}, s_k \\sqrt{|a_k|} v^{\\otimes k}\n\\right\\rangle_{V_k} \\\\\n\u0026amp;= \\sum_{k=0}^{\\infty} s_k |a_k| \\langle u^{\\otimes k}, v^{\\otimes k} \\rangle\\\\\n\u0026amp;= \\sum_{k=0}^{\\infty} a_k \\langle u, v \\rangle^k \\\\\n\u0026amp;= f(\\langle u, v \\rangle).\n\\end{aligned}\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eNote.\u003c/strong\u003e Norm when $u$ is a unit vector: If $\\|u\\| = 1$, then\n$$\n\\|u^{\\otimes k}\\|^2 = \\|u\\|^{2k} = 1.\n$$\u003c/p\u003e\n\u003cp\u003eHence,\n$$\\|\\Phi(u)\\|_\\mathcal{H}^2 = \\sum _{k=0}^{\\infty} |a_k| \\| u^{\\otimes k} \\|^2 = \\sum _{k=0}^{\\infty} |a_k|.$$\u003c/p\u003e\n\u003ch3 id=\"example\"\u003eExample\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003eSine.\u003c/strong\u003e  Let $c \u0026gt; 0$. Then function $ f(x) = \\sin(cx) $ is real analytic\n$$\n\\sin(cx) = cx - \\frac{(cx)^3}{3!} + \\frac{(cx)^5}{5!} - \\ldots\n$$\u003c/p\u003e\n\u003cp\u003eThus, there exists a Hilbert space $\\mathcal{H}$ and transformation $\\Phi, \\Psi: \\mathbb{R}^n \\to \\mathcal{H}$ such that\n$$\n\\langle \\Phi(u), \\Psi(v) \\rangle = \\sin(c \\langle u, v \\rangle)\\quad \\text{for all } u, v \\in \\mathbb{R}^n.\n$$\u003c/p\u003e\n\u003cp\u003eAlso, $\\Phi$ and $\\Psi$ map unit vectors to unit vectors if\n$$\n1 = c + \\frac{c^3}{3!} + \\frac{c^5}{5!} + \\ldots = \\frac{e^c - e^{-c}}{2}.\n$$\u003c/p\u003e\n\u003cp\u003eHence,\n$$\nc = \\ln(1 + \\sqrt{2}).\n$$\u003c/p\u003e\n\u003ch2 id=\"2-kernels-and-feature-maps\"\u003e2. Kernels and feature maps\u003c/h2\u003e\n\u003cp\u003eGiven a function of two variables $ K : \\mathcal{X} \\times \\mathcal{X} \\to \\mathbb{R},$ on some set sets $\\mathcal{X}$, when can we find a Hilbert space $\\mathcal{H}$ and a transformation $\\Phi : \\mathcal{X} \\to \\mathcal{H}$ so that\u003c/p\u003e\n\u003cspan class=\"sidenote\" id=\"note1\"\u003e\n    1. The kernel trick, which expresses a kernel $K(u, v)$ as an inner product, is widely used in machine learning because it lets us handle non-linear models (determined by $K$) with techniques designed for linear models.\n\u003c/span\u003e\n\u003cp\u003e$$\n\\langle \\Phi(u), \\Phi(v) \\rangle = K(u,v)^{\\textcolor{gray} 1}  \\quad \\text{for all } u, v \\in \\mathcal{X}?\n$$\u003c/p\u003e\n\u003cp\u003eThe answer is given by Mercer theorem\u003csup\u003e2\u003c/sup\u003e\n\u003cspan class=\"sidenote\" id=\"note2\"\u003e\n2. \u003ca href=\"https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2023/11/Mercer-original-compressed_compressed.pdf\"\u003e Functions of positive and negative type and their connection with the theory of integral equations\u003c/a\u003e (Mercer, 1909).\n\u003c/span\u003e and, more precisely, Moore-Aronszajn theorem:\u003c/p\u003e\n\u003cp\u003e$-$ The necessary and sufficient condition: $K$ is a \u003cem\u003epositive semidefinite kernel\u003c/em\u003e, i.e., for any $u_1, \\dots, u_N \\in \\mathcal{X}$, the matrix\n$ (K(u_i, u_j))_{i,j=1}^N $ is symmetric and positive semidefinite.\u003c/p\u003e\n\u003cp\u003e$-$ Transformation $\\Phi$ is called a \u003cem\u003efeature map\u003c/em\u003e.\u003c/p\u003e\n\u003cp\u003e$-$ The Hilbert space $\\mathcal{H}$ is called a \u003cem\u003ereproducing kernel Hilbert space\u003c/em\u003e\u003csup\u003e3\u003c/sup\u003e (RKHS).\n\u003cspan class=\"sidenote\" id=\"note3\"\u003e\n3. \u003ca href=\"https://arxiv.org/abs/2106.08443\"\u003e Reproducing Kernel Hilbert Space, Mercer\u0026rsquo;s Theorem, Eigenfunctions, Nyström Method, and Use of Kernels in Machine Learning: Tutorial and Survey\u003c/a\u003e (Ghojogh et al, 2021).\n\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e$-$ Popular positive semidefinite kernels in machine learning include the Gaussian and polynomial kernels given by:\n$$\nK(u,v) = \\exp \\left( -\\frac{\\|u - v\\|^2}{2\\sigma^2} \\right), \\quad K(u,v) = (\\langle u, v \\rangle + r)^k, \\quad u, v \\in \\mathbb{R}^n.\n$$\nwhere $\\sigma \u0026gt; 0, r \u0026gt; 0,$ and $k \\in \\mathbb{N}$ are parameters.\u003c/p\u003e\n\u003ch2 id=\"references\"\u003eReferences\u003c/h2\u003e\n\u003cp\u003e[1] Vershynin, R. (2026). \u003cem\u003eHigh-dimensional probability: An introduction with applications in data science\u003c/em\u003e (2nd ed.). Cambridge University Press.\u003c/p\u003e\n  \u003c/div\u003e\n\u003cp\u003e\n  \u003ca href=\"mailto:nguyencungthanh11@gmail.com?subject=Re: Real%20Analytic%20Function%20in%20Hilbert%20space\" rel='noopener noreferrer'\u003e💬 Reply via e-mail\u003c/a\u003e\n\u003c/p\u003e\n","date_modified":"2026-04-29T00:00:00+00:00","date_published":"2026-04-29T00:00:00+00:00","id":"https://nguyencungthanh.github.io/2026/04/29/real-analytic-function-in-hilbert-space/","summary":"Abstract $\\quad$ We study representations of functions of inner products via embeddings into Hilbert spaces. While polynomials with nonnegative coefficients admit a single feature map realization, this approach fails when negative coefficients are present. To address this limitation, we introduce a two-map framework. Specifically, for any analytic function $f(x) = \\sum_{k=0}^{\\infty} a_k x^k$ with global convergence, we construct a Hilbert space $\\mathcal{H}$ and transformations $\\Phi, \\Psi: \\mathbb{R}^n \\to \\mathcal{H}$ such that $$\\langle \\Phi(u), \\Psi(v) \\rangle = f(\\langle u, v \\rangle) \\quad \\text{for all } u, v \\in \\mathbb{R}^n.$$\n","tags":["Functional Analysis","Tensor","Feature Map"],"title":"Real Analytic Function in Hilbert space","url":"https://nguyencungthanh.github.io/2026/04/29/real-analytic-function-in-hilbert-space/"},{"content_html":"\n  \u003cdiv class='e-content'\u003e\n    \u003ch2 id=\"1-inner-product-spaces-hilbert-spaces\"\u003e1. Inner Product Spaces, Hilbert Spaces\u003c/h2\u003e\n\u003ch3 id=\"11-definition-inner-product-space-hilbert-space\"\u003e1.1 Definition (Inner product space, Hilbert space)\u003c/h3\u003e\n\u003cp\u003e\u0026ndash; An inner product space is a vector space $X$ with an inner product defined on $X$. A Hilbert space is a complete inner product space. An \u003cstrong\u003einner product\u003c/strong\u003e on $X$ is a mapping of $X \\times X$ into the scalar field $K$ of $X$; that is, with every pair of vectors $x$ and $y$ there is associated a scalar which is written $\\langle x,y\\rangle$. For all vectors $x, y, z$ and scalar $\\alpha,$ we have\u003c/p\u003e\n\u003cp\u003e$\n(\\mathrm{IP}1)\\quad \\langle x+y,z\\rangle = \\langle x,z\\rangle + \\langle y,z\\rangle\n$\u003c/p\u003e\n\u003cp\u003e$\n(\\mathrm{IP}2)\\quad \\langle \\alpha x,y\\rangle = \\alpha \\langle x,y\\rangle\n$\u003c/p\u003e\n\u003cp\u003e$\n(\\mathrm{IP}3)\\quad \\langle x,y\\rangle = \\overline{\\langle y,x\\rangle}\n$\u003c/p\u003e\n\u003cp\u003e$\n(\\mathrm{IP}4)\\quad \\langle x,x\\rangle \\ge 0, \\qquad \\langle x,x\\rangle = 0 \\iff x = 0\n$\u003c/p\u003e\n\u003cp\u003eAn inner product on $X$ defines a \u003cem\u003enorm\u003c/em\u003e on $X$ given by\n$$\n\\|x\\| = \\sqrt{\\langle x,x\\rangle}\n$$\u003c/p\u003e\n\u003cp\u003eand a \u003cem\u003emetric\u003c/em\u003e on $X$ given by\n$$\nd(x,y) = \\|x-y\\| = \\sqrt{\\langle x-y, x-y\\rangle}\n$$\u003c/p\u003e\n\u003cp\u003eFrom (IP1) and (IP3), we obtain\u003c/p\u003e\n\u003cp\u003e$\n\\text{(a)}\\quad \\langle \\alpha x + \\beta y,z\\rangle = \\alpha \\langle x,z\\rangle + \\beta \\langle y,z\\rangle\n$\u003c/p\u003e\n\u003cp\u003e$\n\\text{(b)}\\quad \\langle x,\\alpha y\\rangle = \\overline{\\alpha},\\langle x,y\\rangle\n$\u003c/p\u003e\n\u003cp\u003e$\n\\text{(c)}\\quad \\langle x,\\alpha y + \\beta z\\rangle = \\overline{\\alpha},\\langle x,y\\rangle + \\overline{\\beta},\\langle x,z\\rangle\n$\u003c/p\u003e\n\u003cp\u003e\u0026ndash; An inner product space satisfies the parallelogram equality\n$$\n\\|x+y\\|^2 + \\|x-y\\|^2 = 2(\\|x\\|^2 + \\|y\\|^2)\n$$\u003c/p\u003e\n\u003ch3 id=\"12-definition-orthogonality\"\u003e1.2 Definition (Orthogonality)\u003c/h3\u003e\n\u003cp\u003e\u0026ndash; An element $x$ of an inner product space $X$ is said to be orthogonal to an element $y \\in X$ if\n$$\n\\langle x,y\\rangle = 0.\n$$\u003c/p\u003e\n\u003cp\u003e\u0026ndash; We also say that $x$ and $y$ are orthogonal, and we write $x \\perp y$. Similarly, for subsets $A, B \\subset X$ we write $x \\perp A$ if $x \\perp a$ for all $a \\in A$. $A \\perp B$ if $a \\perp b$ for all $a \\in A$ and $b \\in B$.\u003c/p\u003e\n\u003ch3 id=\"examples\"\u003eExamples\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003e1. Euclidean space $\\mathbb{R}^n$.\u003c/strong\u003e Hilbert space with inner product:\n$$\n\\langle x,y\\rangle = \\xi_1 \\eta_1 + \\ldots + \\xi_n \\eta_n\n$$\nwhere $x = (\\xi_i) = (\\xi_1,\\ldots,\\xi_n); \\quad y = (\\eta_i) = (\\eta_1,\\ldots,\\eta_n)$.\nThen\n$$\\|x\\| = \\langle x,x\\rangle^{1/2} = (\\xi_1^2 + \\ldots + \\xi_n^2)^{1/2}$$\nwith Euclidean metric $$\\mathrm{d}(x, y) = \\| x - y \\|| = \\langle x-y, x-y \\rangle^{1/2} = [(\\xi_1 - \\eta_1)^2 + \\ldots +(\\xi_n - \\eta_n)^2]^{1/2}$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2. Unitary space $\\mathbb{C}^n$.\u003c/strong\u003e Hilbert space with inner product:\n$$\n\\langle x,y\\rangle = \\sum_ {i=1}^n \\xi_i \\overline{\\eta_i}\n$$\nThe norm is defined by\n$$\n\\|x\\| = \\left(\\sum_{i=1}^n \\xi_i \\overline{\\xi_i}\\right)^{1/2}\n= \\left(\\sum_{i=1}^n |\\xi_i|^2\\right)^{1/2}\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3. Space $L^2[a,b]$.\u003c/strong\u003e Hilbert space with norm:\n$$\n\\|x\\| = \\left(\\int_a^b x(t)^2\\ \\mathrm{d}t\\right)^{1/2}\n$$\nand can be obtained from the inner product defined by\n$$\n\\langle x,y\\rangle = \\int_a^b x(t)y(t)\\mathrm{d}t\n$$\nConsider complex-valued functions $\\Rightarrow$ complex vector space $\\Rightarrow$ inner product space:\n$$\n\\langle x,y\\rangle = \\int_a^b x(t)\\overline{y(t)}\\mathrm{d}t\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e4. Hilbert sequence space $\\ell^2$.\u003c/strong\u003e Hilbert space with inner product:\n$$\n\\langle x,y\\rangle = \\sum_{j=1}^{\\infty} \\xi_j \\overline{\\eta_j}\n$$\nThe norm is defined by\n$$\n\\|x\\| = \\langle x,x\\rangle^{1/2}\n= \\left(\\sum_{j=1}^{\\infty} |\\xi_j|^2\\right)^{1/2}\n$$\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"2-further-properties-of-inner-product-space\"\u003e2. Further Properties of Inner Product Space\u003c/h2\u003e\n\u003ch3 id=\"21-lemma-schwarz-inequality\"\u003e2.1 Lemma (Schwarz inequality,)\u003c/h3\u003e\n\u003cp\u003e$\\text{(a)}$ We have:\n$$\n\\|\\langle x,y\\rangle\\| \\le \\|x\\|\\ \\|y\\|\n$$\nThe equality sign holds if and only if $\\{x,y\\}$ is a linearly dependent set.\u003c/p\u003e\n\u003cp\u003e$\\text{(b)}$ The norm also satisfies:\n$$\n\\|x+y\\| \\le \\|x\\| + \\|y\\|\n$$\nwhere the equality sign holds if and only if $y = 0$ or $x = cy$ $(c \\in \\mathbb{R},\\ c \\ge 0)$.\u003c/p\u003e\n\u003ch3 id=\"22-lemma-continuity-of-inner-product\"\u003e2.2 Lemma (Continuity of inner product)\u003c/h3\u003e\n\u003cp\u003eIf in an inner product space, $x_n \\to x$ and $y_n \\to y$, then $\\langle x_n,y_n\\rangle \\to \\langle x,y\\rangle$.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e\n\\begin{align}\n|\\langle x_n,y_n\\rangle - \\langle x,y\\rangle| \u0026amp;= |\\langle x_n,y_n\\rangle - \\langle x_n,y\\rangle + \\langle x_n,y\\rangle - \\langle x,y\\rangle| \\\\\n\u0026amp; \\le |\\langle x_n,y_n - y\\rangle| + |\\langle x_n - x,y\\rangle| \\\\\n\u0026amp; \\le \\|x_n\\|\\|y_n - y\\| + \\|x_n - x\\|\\|y\\| \\to 0\n\\end{align}\u003c/p\u003e\n\u003cp\u003eAn isomorphism $T$ of an inner product space $X$ onto an inner product space $\\tilde X$ over the same field is a bijective linear operator $T:X \\to \\tilde X$ which preserves the inner product:\n$$\n\\langle Tx,Ty\\rangle = \\langle x,y\\rangle \\quad \\text{for all } x,y \\in X.\n$$\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e$\\tilde X$ is isomorphic with $X$ $\\Rightarrow$ $X$ and $\\tilde X$ are called isomorphic inner product spaces.\u003c/li\u003e\n\u003cli\u003eBijectivity and linearity guarantees that $T$ is a vector space isomorphism of $X$ onto $\\tilde X$ $\\Rightarrow$ $T$ preserves the whole structure of inner product space.\u003c/li\u003e\n\u003cli\u003e$T$ is also an isometry of $X$ onto $\\tilde X$ because distances in $X$ and $\\tilde X$ are determined by the norms defined by the inner products on $X$ and $\\tilde X$.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"23-theorem-completion\"\u003e2.3 Theorem (Completion)\u003c/h3\u003e\n\u003cp\u003eFor any inner product space $X$ there exists a Hilbert space $H$ and an isomorphism $A$ from $X$ onto a dense subspace $W \\subset H$. The space $H$ is unique except for isomorphisms.\u003c/p\u003e\n\u003cp\u003e\u0026ndash; $H$ is unique except for isometries, that is, two completions $H$ and $\\tilde H$ of $X$ are related by an isometry $T:H \\to \\tilde H$. Then, $T$ must be an isomorphism of the Hilbert space $H$ onto the Hilbert space $\\tilde H$.\u003c/p\u003e\n\u003cp\u003e\u0026ndash; A subspace $Y$ of an inner product space $H$ is defined to be a subspace of $H$, regarded as an inner product space.\u003c/p\u003e\n\u003cp\u003e$$\n!!\\quad Y \\text{ need not to be a Hilbert space because } Y \\text{ may not be complete.}\n$$\u003c/p\u003e\n\u003ch3 id=\"24-theorem-subspace\"\u003e2.4 Theorem (Subspace)\u003c/h3\u003e\n\u003cp\u003eLet $Y$ be a subspace of a Hilbert space $H$. Then:\u003c/p\u003e\n\u003cp\u003e$\\text{(a)} \\quad$ $Y$ is complete if and only if $Y$ is closed in $H$.\u003c/p\u003e\n\u003cp\u003e$\\text{(b)} \\quad$ If $Y$ is finite dimensional, then $Y$ is complete.\u003c/p\u003e\n\u003cp\u003e$\\text{(c)} \\quad$ If $H$ is separable, so is $Y$. More generally, every subset of a separable inner product space is separable.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"3-orthogonal-complements-and-direct-sums\"\u003e3. Orthogonal Complements and Direct Sums\u003c/h2\u003e\n\u003cp\u003eIn a metric space $X$, the distance $\\delta$ from an element $x \\in X$ to a nonempty subset $M \\subset X$ is defined to be:\n$$\n\\delta = \\inf_{\\tilde{y} \\in M} d(x,\\tilde y).\n$$\u003c/p\u003e\n\u003cp\u003eIn a normed space this becomes:\n$$\n\\delta = \\inf_{\\tilde y \\in M} \\|x - \\tilde y\\|.\n$$\u003c/p\u003e\n\u003ch2 id=\"references\"\u003eReferences\u003c/h2\u003e\n\u003cp\u003e[1] Kreyszig, E. (1978). \u003cem\u003eIntroductory functional analysis with applications\u003c/em\u003e. John Wiley \u0026amp; Sons.\u003c/p\u003e\n  \u003c/div\u003e\n\u003cp\u003e\n  \u003ca href=\"mailto:nguyencungthanh11@gmail.com?subject=Re: Hilbert%20Space\" rel='noopener noreferrer'\u003e💬 Reply via e-mail\u003c/a\u003e\n\u003c/p\u003e\n","date_modified":"2026-04-27T00:00:00+00:00","date_published":"2026-04-27T00:00:00+00:00","id":"https://nguyencungthanh.github.io/2026/04/27/hilbert-space/","summary":"","tags":["Functional Analysis","Hilbert Space"],"title":"Hilbert Space","url":"https://nguyencungthanh.github.io/2026/04/27/hilbert-space/"},{"content_html":"\n  \u003cdiv class='e-content'\u003e\n    \u003ch1 id=\"normed-spaces-and-banach-spaces\"\u003eNormed Spaces and Banach Spaces\u003c/h1\u003e\n\u003ch2 id=\"1-definition-normed-space-banach-space\"\u003e1. Definition (Normed Space, Banach Space)\u003c/h2\u003e\n\u003cp\u003eA normed space $X$ is a vector space with a norm defined on it. A Banach space is a complete normed space (complete in the metric defined by the norm). Here, a norm on a (real or complex) vector space $X$ is a real-valued function on $X$ whose value at an $x \\in X$ is denoted by $\\| x \\|$ (real \u0026ldquo;norm of $X$\u0026rdquo;) and which has the properties:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e$\\|x\\| \\ge 0$\u003c/li\u003e\n\u003cli\u003e$\\|x\\| = 0 \\iff x = 0$\u003c/li\u003e\n\u003cli\u003e$\\|\\alpha x\\| = |\\alpha|\\|x\\|$\u003c/li\u003e\n\u003cli\u003e$\\|x + y\\| \\le \\|x\\| + \\|y\\|$ $\\quad$ (Triangle inequality)\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eA norm on $X$ defines a metric $d$ on $X$ by\n$$\nd(x,y) = \\|x - y\\|,\n$$\ncalled the \u003cem\u003emetric induced\u003c/em\u003e by the norm. The normed space thus defined is denoted by $(X, \\| \\cdot\\|)$, or simply $X$.\u003c/p\u003e\n\u003ch3 id=\"example\"\u003eExample\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003e1. Euclidean space  $\\mathbb{R}^n$ and unitary space $\\mathbb{C}^n$\u003c/strong\u003e.\nBanach space with norm\n$$\n\\|x\\| = \\left( \\sum_{i=1}^{n} |x_i|^2 \\right)^{\\frac{1}{2}}\n$$\n$\\mathbb{R}^n$ and $\\mathbb{C}^n$ are complete and the induced metric is\n$$\nd(x,y) = \\|x - y\\| = \\left( \\sum_{i=1}^{n} |x_i - y_i|^2 \\right)^{\\frac{1}{2}}.\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2. Space $\\ell^p$.\u003c/strong\u003e\nBanach space with norm\u003cbr\u003e\n$$ \\|x\\| = \\left( \\sum_{i = 1}^{\\infty} |x_i|^p \\right)^{\\frac{1}{p}} $$\nand the induced metric is\n$$\nd(x, y) = \\|x - y\\| = \\left( \\sum_{i=1}^{n} |x_i - y_i|^p \\right)^{\\frac{1}{p}}.\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3. Space $\\ell^{\\infty}$.\u003c/strong\u003e\nBanach space with norm\u003cbr\u003e\n$$\n\\|x\\|_\\infty = \\sup _{i}\\|x_i\\|\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e4 Space $C[a, b]$.\u003c/strong\u003e\nBanach space with norm\u003cbr\u003e\n$$\n\\|x\\| = \\max _{t \\in J}\\|x(t)\\|\n$$\nwhere $J = [a,b]$.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e5. Incomplete normed space.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e$d(x, y) = \\int _{0}^{1} | x(t) - y(t) | \\mathrm{d}t$ induce the norm\n$$\\int _{0}^{1} |x(t)| \\mathrm{d}t$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e6. Incomplete normed space and its completion $L^p[a,b]$.\u003c/strong\u003e\nThe Banach space $L^p[a,b]$ with norm\n$$ \\|x\\| _p = \\left( \\int _{a}^{b} |x(t)| \\mathrm{d}t  \\right)^{\\frac{1}{p}} $$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e7. Space $s$.\u003c/strong\u003e\nMetric defined by $$d(x,y) = \\sum _{j = 1}^{\\infty} \\dfrac{1}{2^j} \\dfrac{|x_j-y_j|}{1 + |x_j-y_j|}$$\ncannot be obtained a norm\u003c/p\u003e\n\u003ch3 id=\"11-lemma-translation-invariance\"\u003e1.1 Lemma (Translation invariance)\u003c/h3\u003e\n\u003cp\u003eA metric $d$ induced by a norm on a normed space $X$ satisfies\u003c/p\u003e\n\u003cp\u003e$\\text{(a)} \\quad$ $d(x+a,y+a) = d(x,y)$\u003c/p\u003e\n\u003cp\u003e$\\text{(b)} \\quad$ $d(\\alpha x, \\alpha y) = |\\alpha| d(x,y)$\u003c/p\u003e\n\u003cp\u003efor all $x,y \\in X$ and every scalar $\\alpha$.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"2-further-properties-of-normed-space\"\u003e2. Further properties of normed space\u003c/h2\u003e\n\u003ch3 id=\"21-theorem-subspace-of-banach-space\"\u003e2.1 Theorem (Subspace of Banach space)\u003c/h3\u003e\n\u003cp\u003eA subspace $Y$ of a Banach space $X$ is complete if and only if $Y$ is closed in $X$.\u003c/p\u003e\n\u003cp\u003e\u0026ndash; \u003cstrong\u003eConvergence of sequences\u003c/strong\u003e\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eA sequence $(x_n)$ in a normed space $X$ is convergent if $X$ contains an $x$ such that\n$$\\lim _{n \\to \\infty} \\|x_n - x \\| = 0.$$\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThen we write $x_n \\to x$ and call $x$ the limit of $(x_n)$.\u003c/p\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003eA sequence $(x_n)$ in a normed space $X$ is Cauchy if for every $\\epsilon \u0026gt; 0$ there is an $N$ such that $$\\|x_m - x_n\\| \u0026lt; \\epsilon \\quad \\forall m, n \u0026gt; N$$\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u0026ndash; \u003cstrong\u003eInfinite series\u003c/strong\u003e $(x_k)$ is a sequence in a normed space $X$, we can associate with $(x_k)$ the sequence $(s_n)$ of partial sums $$s_n = x_1 + x_2 + \\ldots + x_n.$$\u003c/p\u003e\n\u003cp\u003eIf $(s_n)$ is convergent,\n$\ns_n \\to s,\n$\nor $\\|s_n - s\\| \\to 0$ then the infinite series $\\sum_{k = 1}^\\infty x_k$ is said to converge, $s$ is called the sum of the series and we write\n$$\ns = \\sum_{k=1}^{\\infty} x_k = x_1 + x_2 + \\ldots.\n$$\u003c/p\u003e\n\u003cp\u003eIf\n$\n\\sum_{k=1}^{\\infty} \\|x_k\\|\n$\nconverges, then the series is said to be \u003cem\u003eabsolutely convergent\u003c/em\u003e.\u003c/p\u003e\n\u003cp\u003e\u0026ndash; \u003cstrong\u003eBasis.\u003c/strong\u003e A normed space $X$ contains a sequence $(e_n)$ with the property that for every $x \\in X$ here is a unique sequence of scalars $(\\alpha_n)$ such that\n$$\n\\|x - \\sum_{k=1}^{n} \\alpha_k e_k\\| \\to 0 \\quad \\text{ (as } n \\to \\infty)\n$$\nthen $(e_n)$ is called a \u003cstrong\u003eSchander basis\u003c/strong\u003e for $X$. The series $\\sum_{k=1}^{\\infty} \\alpha_k e_k$ which has the sum $x$ is then called the expansion of $x$ with to $(e_n)$, and we write\n$$x = \\sum_{k=1}^{\\infty} \\alpha_k e_k$$\u003c/p\u003e\n\u003ch4 id=\"example-1\"\u003eExample\u003c/h4\u003e\n\u003cp\u003e$\\ell^p$ has a Schauder, namely $(e_n),$ where $e_n = (\\delta_{nj})$, that is, $e_n$ is the sequence whose $n^{\\text{th}}$ term is 1 and all other term are zero; thus\n$$\ne_1 = (1,0,0,\\dots), \\quad\ne_2 = (0,1,0,\\dots), \\quad\ne_3 = (0,0,1,\\dots), \\ \\text{etc.}\n$$\u003c/p\u003e\n\u003ch3 id=\"21-theorem--completion\"\u003e2.1 Theorem  (Completion)\u003c/h3\u003e\n\u003cp\u003eLet $X = (X, \\|\\cdot\\|)$ be a normed space. Then there exists a Banach space $\\tilde{X}$ and an isometry $A$ from $X$ onto a subspace $W$ of $\\tilde{X}$ which is dense in $\\tilde{X}$. The space $\\tilde{X}$ is unique, except for isometris.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"3-finite-dimensional-normed-spaces-and-subspaces\"\u003e3. Finite Dimensional Normed Spaces and Subspaces.\u003c/h2\u003e\n\u003ch3 id=\"31-lemma-linear-combination\"\u003e3.1 Lemma (Linear combination)\u003c/h3\u003e\n\u003cp\u003eLet $\\{ x_1, \\ldots, x_n \\}$ be a linearly independent set of vectors in a normed space $X$ (of any dimension). Then there is a number $c \u0026gt; 0$ such that for every choice of scalars $\\alpha_1, \\ldots, \\alpha_n$ we have\u003cbr\u003e\n\\begin{equation}\n\\| \\alpha_1x_1 + \\ldots + \\alpha_n x_n \\| \\ge c(| \\alpha_1 | + \\ldots + |\\alpha_n|) \\tag{1}\n\\end{equation}\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eWe write $S = |a_1| + \\cdots + |a_n|$. If $S = 0$, all $\\alpha_j$ are zero, so (1) holds for any $c$. Let $S \u0026gt; 0$. Then (1) is equivalent to the inequality which we obtain from (1) by dividing $S$ and writing $\\beta_j = \\dfrac{\\alpha_j}{S}$, that is\u003c/p\u003e\n\u003cp\u003e\\begin{equation}\n\\left\\| \\beta_1 x_1 + \\cdots + \\beta_n x_n \\right\\| \\ge c\n\\qquad \\left( \\sum_{j=1}^n |\\beta_j| = 1 \\right)  \\tag{2}\n\\end{equation}\u003c/p\u003e\n\u003cp\u003eSuppose for the contradiction, there exists a sequence $(y_m)$ of vectors\u003c/p\u003e\n\u003cp\u003e$$\ny_m = \\beta_1^{(m)} x_1 + \\cdots + \\beta_n^{(m)} x_n\n$$\u003c/p\u003e\n\u003cp\u003esuch that\n$\n|y_m| \\to 0 \\quad \\text{as } m \\to \\infty.\n$\u003c/p\u003e\n\u003cp\u003eSince\n$\n\\sum_{j=1}^n \\left|\\beta_j^{(m)}\\right| = 1,\n$\nwe have $\\left|\\beta_j^{(m)}\\right| \\le 1$. Hence for each $j$, the sequence\n$$\n\\left(\\beta_j^{(m)}\\right) = \\left(\\beta_j^{(1)}, \\beta_j^{(2)}, \\dots\\right)\n$$\nis bounded. Consequently, by the Bolzano-Weierstrass theorem, $\\left(\\beta_j^{(m)}\\right)$ has a convergent subsequence. Let $\\beta_1$ denote the limit of that subsequence, let $(y_{1,m})$ denote the corresponding subsequence of $(y_m)$. By the same argument, $(y_{1,m})$ has a subsequence $(y_{2,m})$ for which the corresponding subsequence of scalars $\\beta_j^{(m)}$ converges; let $\\beta_2$ denote the limit. Continuing in this way, after $n$ steps we obtain a subsequence\u003c/p\u003e\n\u003cp\u003e$$\n(y_{n,m}) = (y_{n,1}, y_{n,2}, \\dots)\n$$\u003c/p\u003e\n\u003cp\u003eof $(y_m)$ whose terms are of the form\u003c/p\u003e\n\u003cp\u003e$$\ny_{n,m} = \\sum_{j=1}^n \\gamma_j^{(m)} x_j.\n$$\u003c/p\u003e\n\u003cp\u003ewith scalars $\\gamma_j^{(m)}$ satisfying\n$\n\\gamma_j^{(m)} \\to \\beta_j \\text{ as } m \\to \\infty.\n$\nHence, as $m \\to \\infty$,\n$\ny_{n,m} \\to y = \\sum_{j=1}^n \\beta_j x_j\n$\u003c/p\u003e\n\u003cp\u003ewhere\n$\n\\sum_{j=1}^n |\\beta_j| = 1,\n$\nso that not all $\\beta_j$ can be zero. Since $\\{x_1,\\dots,x_n\\}$ is a linearly independent then $y \\ne 0$.\u003c/p\u003e\n\u003cp\u003eOn the other hand, $y_{n,m} \\to y$ implies $\\|y_{n,m}\\| \\to \\|y\\|$. Since $\\|y_m\\| \\to 0$ by assumption and $(y_{n,m})$ is a subsequence of $(y_m)$, we must have $\\|y_{n,m}\\| \\to 0$. Hence $\\|y\\| = 0$ so that $y = 0$ (contradiction).\u003c/p\u003e\n\u003ch3 id=\"32-theorem-completeness\"\u003e3.2 Theorem (Completeness)\u003c/h3\u003e\n\u003cp\u003eEvery finite dimensional subspace $Y$ of a normed space $X$ is complete. In particular, every finite dimensional normed space is complete.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eConsider an arbitrary Cauchy sequence $(y_m)$ in $Y$ and show that it is convergent in $Y$; the limit will be denoted by $y$.\u003c/p\u003e\n\u003cp\u003eLet $\\dim Y = n$ and $\\{e_1,\\ldots,e_n\\}$ any basis for $Y$. Then each $y_m$ has a unique representation of the form\u003c/p\u003e\n\u003cp\u003e$$\ny_m = \\alpha_1^{(m)} e_1 + \\cdots + \\alpha_n^{(m)} e_n.\n$$\u003c/p\u003e\n\u003cp\u003eSince $(y_m)$ is a Cauchy sequence, for every $\\varepsilon \u0026gt; 0$ there is an $N$ such that $\\|y_m - y_r\\| \u0026lt; \\varepsilon$ when $m,r \u0026gt; N$. We have for some $c \u0026gt; 0$\u003c/p\u003e\n\u003cp\u003e$$\n\\varepsilon \u0026gt; \\|y_m - y_r\\|\n= \\left\\| \\sum_{j=1}^n \\left(\\alpha_j^{(m)} - \\alpha_j^{(r)}\\right)e_j \\right\\|\n\\ge c \\sum_{j=1}^n \\left|\\alpha_j^{(m)} - \\alpha_j^{(r)}\\right|\n$$\u003c/p\u003e\n\u003cp\u003e$$\n\\Rightarrow \\left|a_j^{(m)} - a_j^{(r)}\\right| \\le \\frac{\\varepsilon}{c}.\n$$\u003c/p\u003e\n\u003cp\u003eThis shows that each of the $n$ sequences\u003c/p\u003e\n\u003cp\u003e$$\n\\left(\\alpha_j^{(m)}\\right)=\\left(\\alpha_j^{(1)},\\alpha_j^{(2)},\\ldots\\right), \\qquad j=1,\\ldots,n\n$$\u003c/p\u003e\n\u003cp\u003eis Cauchy in $\\mathbb{R}$ or $\\mathbb{C}$. Hence it converges; let $\\alpha_j$ denote the limit. Using these $n$ limits $a_1,\\ldots,a_n$, we define\u003c/p\u003e\n\u003cp\u003e$$\ny=\\alpha_1e_1+\\cdots+\\alpha_ne_n.\n$$\u003c/p\u003e\n\u003cp\u003eClearly, $y\\in Y$ and\u003c/p\u003e\n\u003cp\u003e$$\n\\|y_m-y\\| =\n\\left\\|\\sum_{j=1}^n \\left(\\alpha_j^{(m)}-\\alpha_j\\right)e_j\\right\\|\n\\le\n\\sum_{j=1}^n \\left|\\alpha_j^{(m)}-\\alpha_j\\right|\\|e_j\\|.\n$$\u003c/p\u003e\n\u003cp\u003eOn the right, $\\alpha_j^{(m)}\\to \\alpha_j$. Hence $\\|y_m-y\\|\\to 0$, that is $y_m\\to y$.\u003c/p\u003e\n\u003cp\u003eThis shows that $(y_m)$ is convergent in $Y$. Since $(y_m)$ was an arbitrary Cauchy sequence in $Y$, this proves that $Y$ is complete.\u003c/p\u003e\n\u003ch3 id=\"33-theorem-closedness\"\u003e3.3 Theorem (Closedness)\u003c/h3\u003e\n\u003cp\u003eEvery finite dimensional subspace $Y$ of a normed space $X$ is closed in $X$.\u003c/p\u003e\n\u003ch3 id=\"34-definition-equivalent-norms\"\u003e3.4 Definition (Equivalent norms)\u003c/h3\u003e\n\u003cp\u003eA norm $\\|\\cdot\\|$ on a vector space $X$ is said to be equivalent to a norm $|\\cdot|_0$ on $X$ if there are positive numbers $a$ and $b$ such that for all $x\\in X$ we have\u003c/p\u003e\n\u003cp\u003e$$\na\\|x\\|_0 \\le \\|x\\| \\le b\\|x\\|_0.\n$$\u003c/p\u003e\n\u003cp\u003eEquivalent norms on $X$ define the same topology for $X$.\u003c/p\u003e\n\u003ch3 id=\"35-theorem-equivalent-norms\"\u003e3.5 Theorem (Equivalent norms)\u003c/h3\u003e\n\u003cp\u003eOn a finite dimensional vector space $X$ any norm $\\|\\cdot\\|$ is equivalent to any other norm $\\|\\cdot\\|_0$.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eLet $\\dim X=n$ and ${e_1,\\ldots,e_n}$ any basis for $X$. Then every $x\\in X$ has a unique representation\u003c/p\u003e\n\u003cp\u003e$$\nx=\\alpha_1e_1+\\cdots+\\alpha_ne_n.\n$$\u003c/p\u003e\n\u003cp\u003eThen there exists $c\u0026gt;0$ such that\u003c/p\u003e\n\u003cp\u003e$$\n\\|x\\|\\ge c\\left(|a_1|+\\cdots+|a_n|\\right).\n$$\u003c/p\u003e\n\u003cp\u003eOn the other hand, the triangle inequality gives\u003c/p\u003e\n\u003cp\u003e$$\n\\|x\\|_0 \\le \\sum _{j = 1}^n \\left|\\alpha_j \\right|\\|e_j\\|_0 \\le k \\sum _{j=1}^n|\\alpha_j| \\qquad k = \\max_j \\|e_j\\|_0.\n$$\u003c/p\u003e\n\u003cp\u003eHence $a\\|x\\|_0 \\le \\|x\\|,$ where $ a=\\frac{c}{k}\u0026gt;0.$\nInterchange of the roles of $\\|\\cdot\\|$ and $\\|\\cdot\\|_0$, we also have\n$\n\\|x\\|\\le b\\|x\\|_0,\n$\nso these norms are equivalent.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"4-compactness-and-finite-dimension\"\u003e4. Compactness and Finite Dimension\u003c/h2\u003e\n\u003ch3 id=\"41-definition-compactness\"\u003e4.1 Definition (Compactness)\u003c/h3\u003e\n\u003cp\u003eA metric space $X$ is said to be compact if every sequence in $X$ has a convergent subsequence. A subset $M$ of $X$ is said to be compact if $M$ is compact considered as a subspace of $X$, that is, if every sequence in $M$ has a convergent subsequence whose limit is an element of $M$.\u003c/p\u003e\n\u003ch3 id=\"42-lemma-compactness\"\u003e4.2 Lemma (Compactness)\u003c/h3\u003e\n\u003cp\u003eA compact subset $M$ of a metric space is closed and bounded.\u003c/p\u003e\n\u003ch3 id=\"43-theorem-compactness\"\u003e4.3 Theorem (Compactness)\u003c/h3\u003e\n\u003cp\u003eIn a finite dimensional normed space $X$, any subset $M \\subset X$ is compact if and only if $M$ is closed and bounded.\u003c/p\u003e\n\u003ch3 id=\"44-f-rieszs-lemma\"\u003e4.4 F. Riesz\u0026rsquo;s Lemma\u003c/h3\u003e\n\u003cp\u003eLet $Y$ and $Z$ be subspaces of a normed space $X$ (of any dimension), and suppose that $Y$ is closed and is a proper subset of $Z$. Then for every real number $\\theta$ in the interval $(0,1)$ there is a $z \\in Z$ such that\u003c/p\u003e\n\u003cp\u003e$$\n\\|z\\| = 1, \\qquad \\|z - y\\| \\ge \\theta \\quad \\text{for all } y \\in Y.\n$$\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eWe consider any $v \\in Z \\setminus Y$ and denote its distance from $Y$ by $a$, that is,\u003c/p\u003e\n\u003cp\u003e$$\na = \\inf_{y \\in Y} |v - y|.\n$$\u003c/p\u003e\n\u003cp\u003eClearly, $a \u0026gt; 0$ since $Y$ is closed. We now take any $\\theta \\in (0,1)$. By the definition of an infimum there is a $y_0 \\in Y$ such that\u003c/p\u003e\n\u003cp\u003e$$\na \\le |v - y_0| \\le \\frac{a}{\\theta}.\n$$\u003c/p\u003e\n\u003cp\u003eLet\n$\nz = c(v - y_0), \\text{ where } c = \\frac{1}{|v - y_0|},\n$\nthen $\\|z\\| = 1$, and we show that $\\|z - y\\| \\ge \\theta$ for every $y \\in Y$. We have\u003c/p\u003e\n\u003cp\u003e\\begin{align*}\n\\|z - y\\| \u0026amp;= |c(v - y_0) - y| \\\\\n\u0026amp;= c \\|v - y_0 - c^{-1} y\\| \\\\\n\u0026amp;= c \\| v - y_1 \\| \\qquad (y_1 = y_0 + c^{-1} y)\n\\end{align*}\u003c/p\u003e\n\u003cp\u003eThe form of $y_1$ show that $y_1 \\in Y$. Hence $\\|v - y_1\\| \\ge a$ and we obtain\n$$\n\\|z - y\\| = c \\|v - y_1\\| \\ge ca = \\frac{a}{|v - y_0|} \\ge \\frac{a}{a/\\theta} = \\theta.\n$$\u003c/p\u003e\n\u003ch3 id=\"45-theorem-finite-dimension\"\u003e4.5 Theorem (Finite dimension)\u003c/h3\u003e\n\u003cp\u003eIf a normed space $X$ has the property that the closed unit ball $M = \\{x \\mid \\|x\\| \\le 1\\}$ is compact, then $X$ is finite dimensional.\u003c/p\u003e\n\u003ch3 id=\"46-theorem-continuous-mapping\"\u003e4.6 Theorem (Continuous mapping)\u003c/h3\u003e\n\u003cp\u003eLet $X$ and $Y$ be metric spaces and $T : X \\to Y$ a continuous mapping. Then the image of a compact subset $M$ of $X$ under $T$ is compact.\u003c/p\u003e\n\u003ch3 id=\"47-corollary-maximum-and-minimum\"\u003e4.7 Corollary (Maximum and Minimum)\u003c/h3\u003e\n\u003cp\u003eA continuous mapping $T$ of a compact subset $M$ of a metric space $X$ into $\\mathbb{R}$ assumes a maximum and minimum at some points of $M$.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"5-linear-operators\"\u003e5. Linear operators\u003c/h2\u003e\n\u003ch3 id=\"51-definition-linear-operator\"\u003e5.1 Definition (Linear operator)\u003c/h3\u003e\n\u003cp\u003eA linear operator $T$ is an operator such that\u003c/p\u003e\n\u003cp\u003e$\\text{(i)} \\quad$ The domain $\\mathcal{D}(T)$ of $T$ is a vector space and the range $\\mathcal{R}(T)$ lies in a vector space over the same field.\u003c/p\u003e\n\u003cp\u003e$\\text{(ii)} \\quad$ For all $x,y \\in \\mathcal{D}(T)$ and scalars $\\alpha$,\n$$\nT(x+y) = Tx + Ty\n$$\n$$\nT(\\alpha x) = \\alpha Tx\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eNote.\u003c/strong\u003e $\\mathcal{N}(T)$ denotes the null space of $T$, the set of all $x \\in \\mathcal{D}(T)$ such that $Tx = 0$ (Kernel).\u003c/p\u003e\n\u003cp\u003e$T$ is a homomorphism of a vector space (its domain) into another space, that is $T$ preserves the two operations of vector space.\u003c/p\u003e\n\u003ch3 id=\"example-2\"\u003eExample\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003e1. Identity operator.\u003c/strong\u003e The identity operator $I_X : X \\to X$ is defined by $I_X x = x$ for all $x \\in X$. Write simply $I$ for $I_X$, thus $Ix = x$.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2. Differentiation.\u003c/strong\u003e Let $X$ be the vector space of all polynomials on $[a,b]$. We may define a linear operator $T$ on $X$ by setting\n$$\nT x(t) = x\u0026rsquo;(t)\n$$\nfor every $x \\in X$.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3. Integration.\u003c/strong\u003e A linear operator $T$ from $C[a,b]$ into itself can be defined by\n$$\nT x(t) = \\int_a^t x(\\tau)\\ \\mathrm{d}\\tau\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e4. Multiplication by $t$.\u003c/strong\u003e Another linear operator from $C[a,b]$ into itself is defined by\n$$\nT x(t) = t x(t)\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e5. Elementary vector algebra.\u003c/strong\u003e Cross product with one factor kept fixed defines a linear operator $T_1 : \\mathbb{R}^3 \\to \\mathbb{R}^3$. Similarly, the dot product with one fixed factor defines a linear operator $T_2 : \\mathbb{R}^3 \\to \\mathbb{R}$, say\n$$\nT_2 x = x \\cdot a = \\xi_1 \\alpha_1 + \\xi_2 \\alpha_2 + \\xi_3 \\alpha_3\n$$\nwhere $\\alpha = (\\alpha_j) \\in \\mathbb{R}^3$ is fixed.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e6. Matrices.\u003c/strong\u003e A real matrix $A = (\\alpha_{jk})$ with $r$ rows and $n$ columns defines an operator $T : \\mathbb{R}^n \\to \\mathbb{R}^r$ by means of\n$$\ny = Ax\n$$\nwhere $x = (x_j)$ has $n$ components and $y = (y_j)$ has $r$ components:\u003c/p\u003e\n\u003cp\u003e$$\n\\begin{bmatrix}\ny_1 \\\\\ny_2 \\\\\n\\vdots \\\\\ny_r\n\\end{bmatrix} =\n\\begin{bmatrix}\n\\alpha_{11} \u0026amp; \\alpha_{12} \u0026amp; \\cdots \u0026amp; \\alpha_{1n} \\\\\n\\alpha_{21} \u0026amp; \\alpha_{22} \u0026amp; \\cdots \u0026amp; \\alpha_{2n} \\\\\n\\vdots \u0026amp; \\vdots \u0026amp; \\ddots \u0026amp; \\vdots \\\\\n\\alpha_{r1} \u0026amp; \\alpha_{r2} \u0026amp; \\cdots \u0026amp; \\alpha_{rn}\n\\end{bmatrix}\n\\begin{bmatrix}\nx_1 \\\\\nx_2 \\\\\n\\vdots \\\\\nx_n\n\\end{bmatrix}\n$$\u003c/p\u003e\n\u003ch3 id=\"52-theorem-range-and-null-space\"\u003e5.2 Theorem (Range and null space)\u003c/h3\u003e\n\u003cp\u003eLet $T$ be a linear operator, then\u003c/p\u003e\n\u003cp\u003e$\\text{(a)} \\quad$ The range $\\mathcal{R}(T)$ is a vector space.\u003c/p\u003e\n\u003cp\u003e$\\text{(b)} \\quad$ If $\\dim \\mathcal{D}(T) = n \u0026lt; \\infty$, then $\\dim \\mathcal{R}(T) \\le n$.\u003c/p\u003e\n\u003cp\u003e$\\text{(c)} \\quad$ The null space $\\mathcal{N}(T)$ is a vector space.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eInverse of linear operator.\u003c/strong\u003e $T : \\mathcal{D}(T) \\to Y$ is injective or one-to-one, there exists the mapping\u003c/p\u003e\n\u003cp\u003e\\begin{align}\nT^{-1} : \u0026amp; \\mathcal{R}(T) \\to \\mathcal{D}(T), \\\\\n\u0026amp; \\quad y_0 \\text{ }\\text{ } \\mapsto \\text{ } \\text{ } x_0 \\quad (y_0 = Tx_0)\n\\end{align}\nWe have\n$$\nT^{-1}Tx = x, \\qquad TT^{-1}y = y\n$$\u003c/p\u003e\n\u003ch3 id=\"53-theorem-inverse-operator\"\u003e5.3 Theorem (Inverse operator)\u003c/h3\u003e\n\u003cp\u003eLet $X, Y$ be vector spaces, both real or complex. Let $T : \\mathcal{D}(T) \\to Y$ be a linear operator with domain $\\mathcal{D}(T) \\subset X$ and $\\mathcal{R}(T) \\subset Y$. Then:\u003c/p\u003e\n\u003cp\u003e$\\text{(a)} \\quad$ The inverse $T^{-1} : \\mathcal{R}(T) \\to \\mathcal{D}(T)$ exists if and only if\u003c/p\u003e\n\u003cp\u003e$$\nTx = 0 \\Rightarrow x = 0\n$$\u003c/p\u003e\n\u003cp\u003e$\\text{(b)} \\quad$ If $T^{-1}$ exists, it is a linear operator.\u003c/p\u003e\n\u003cp\u003e$\\text{(c)} \\quad$ If $\\dim \\mathcal{D}(T) = n \u0026lt; \\infty$ and $T^{-1}$ exists, then\n$\n\\dim \\mathcal{R}(T) = \\dim \\mathcal{D}(T)\n$\u003c/p\u003e\n\u003ch3 id=\"54-lemma-inverse-of-product\"\u003e5.4 Lemma (Inverse of product)\u003c/h3\u003e\n\u003cp\u003eLet $T : X \\to Y$ and $S : Y \\to Z$ be bijective linear operators, where $X, Y, Z$ are vector spaces. Then the inverse $(ST)^{-1} : Z \\to X$ of the product $ST$ exists, and\u003c/p\u003e\n\u003cp\u003e$$\n(ST)^{-1} = T^{-1} S^{-1}.\n$$\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"6-bounded-and-continuous-linear-operators\"\u003e6. Bounded and Continuous Linear Operators\u003c/h2\u003e\n\u003ch3 id=\"61-definition-bounded-linear-operator\"\u003e6.1 Definition (Bounded linear operator)\u003c/h3\u003e\n\u003cp\u003eLet $X$ and $Y$ be normed spaces and $T : \\mathcal{D}(T) \\to Y$ a linear operator, where $\\mathcal{D}(T) \\subset X$. The operator $T$ is said to be bounded if there is a real number $c$ such that for all $x \\in \\mathcal{D}(T)$,\u003c/p\u003e\n\u003cp\u003e$$\n\\|Tx\\| \\le c \\|x\\|\n$$\u003c/p\u003e\n\u003cp\u003e$\\Rightarrow$ $c$ is at least as big as the supremum of the expression on the left taken over $\\mathcal{D}(T) - \\{0\\}$. This supremum is denoted by $\\|T\\|$; thus\u003c/p\u003e\n\u003cp\u003e$$\n\\|T\\| = \\sup_{\\substack{x \\in \\mathcal{D}(T) \\\\ x \\ne 0}} \\frac{\\|Tx\\|}{\\|x\\|}\n$$\u003c/p\u003e\n\u003cp\u003e$\\|T\\|$ is called the norm of the operator $T$. If $\\mathcal{D}(T) = \\{0\\}$ we define $\\|T\\| = 0$. Note that, with $c = \\|T\\|$,\u003c/p\u003e\n\u003cp\u003e$$\n\\|Tx\\| \\le \\|T\\| \\|x\\|\n$$\u003c/p\u003e\n\u003ch3 id=\"62-lemma-norm\"\u003e6.2 Lemma (Norm)\u003c/h3\u003e\n\u003cp\u003eLet $T$ be a bounded linear operator. Then, an alternative formula for the norm of $T$ is\u003c/p\u003e\n\u003cp\u003e$$\n\\|T\\| = \\sup_{\\substack{x \\in \\mathcal{D}(T) \\\\ \\|x\\| = 1}} \\|Tx\\|\n$$\u003c/p\u003e\n\u003ch3 id=\"example-3\"\u003eExample\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003e1. Identity operator.\u003c/strong\u003e The identity operator $I : X \\to X$ on a normed space $X \\ne \\{0\\}$ is bounded and has norm $\\|I\\| = 1$.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2. Differentiation operator.\u003c/strong\u003e Let $X$ be the normed space of all polynomials on $J = [0,1]$ with norm given $\\|x\\| = \\max |x(t)|$, $t \\in J$. A differentiation operator $T$ is defined on $X$ by\n$$\nTx(t) = x\u0026rsquo;(t)\n$$\nThis operator is linear but not bounded.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3. Integral operator.\u003c/strong\u003e  Define an integral operator\n$\nT: C[0,1] \\to C[0,1]\n$\nby\n$$\ny = Tx, \\text{ where} \\quad  y(t)=\\int_0^1 k(t,\\tau)x(\\tau) \\mathrm{d}\\tau.\n$$\nHere $k$ is a given function, which is called the \u003cem\u003ekernel\u003c/em\u003e of $T$, and it is assumed to be continuous on the closed square\n$\nG = J \\times J\n$\nin the $t\\tau$-plane, where\n$\nJ=[0,1].\n$\u003c/p\u003e\n\u003cp\u003eThis operator is linear, $T$ is bounded.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e4. Matrix.\u003c/strong\u003e A real matrix\n$\nA=(\\alpha_{jk})\n$\nwith $r$ rows and $n$ columns defines an operator\n$\nT:\\mathbb{R}^n \\to \\mathbb{R}^r\n$\nby means of\n$$\ny=Ax.\n$$\nwhere\n$\nx=(x_i) \\text{ and } y=(y_i); \\quad y_i=\\sum_{k=1}^n \\alpha_{jk}x_k \\quad (j=\\overline{1,r}).\n$\n$T$ is linear and bounded.\u003c/p\u003e\n\u003ch3 id=\"63-theorem-finite-dimension\"\u003e6.3 Theorem (Finite dimension)\u003c/h3\u003e\n\u003cp\u003eIf a normed space $X$ is finite dimensional, then every linear operator on $X$ is bounded.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e\nLet $\\dim X=n$ and $\\{e_1,\\dots,e_n\\}$ be a basis for $X$. We take any\n$\nx=\\sum \\xi_i e_i\n$\nand consider any linear operator $T$ on $X$. Since $T$ is linear,\n\\begin{align}\n\\|Tx\\|=\\left\\| \\sum \\xi_i Te_i \\right\\| \u0026amp; \\le \\sum |\\xi_i|\\ \\|Te_i\\| \\\\\n\u0026amp; \\le \\max_k \\|Te_k\\| \\sum |\\xi_i| \\\\\n\u0026amp; \\le \\max_k \\|Te_k\\| \\frac{1}{c}\\left\\|\\sum \\xi_i e_i\\right\\|\n= \\delta \\|x\\|.\n\\end{align}\nwhere\n$\n\\delta=\\frac{1}{c}\\max\\limits_k \\|Te_k\\|.\n$\u003c/p\u003e\n\u003ch3 id=\"64-theorem-continuity-and-boundedness\"\u003e6.4 Theorem (Continuity and boundedness)\u003c/h3\u003e\n\u003cp\u003eLet\n$\nT:\\mathcal{D}(T)\\to Y\n$\nbe a linear operator, where $\\mathcal{D}(T)\\subset X$ and $X,Y$ are normed spaces. Then:\u003c/p\u003e\n\u003cp\u003e$\\text{(a)} \\quad$ $T$ is continuous if and only if $T$ is bounded.\u003c/p\u003e\n\u003cp\u003e$\\text{(b)} \\quad$ If $T$ is continuous at a single point, it is continuous.\u003c/p\u003e\n\u003ch3 id=\"65-corollary-continuity-null-space\"\u003e6.5 Corollary (Continuity, null space)\u003c/h3\u003e\n\u003cp\u003eLet $T$ be a bounded linear operator. Then:\u003c/p\u003e\n\u003cp\u003e$\\text{(a)} \\quad$ $x_n\\to x$ $( \\text{where } x_n,x\\in \\mathcal{D}(T))$ implies $Tx_n\\to Tx$.\u003c/p\u003e\n\u003cp\u003e$\\text{(b)} \\quad$ The null space $\\mathcal{N}(T)$ is closed.\u003c/p\u003e\n\u003ch3 id=\"66-theorem-bounded-linear-extension\"\u003e6.6 Theorem (Bounded linear extension)\u003c/h3\u003e\n\u003cp\u003eLet\n$\nT:\\mathcal{D}(T)\\to Y\n$\nbe a bounded linear operator, where $\\mathcal{D}(T)$ lies in a normed space $X$ and $Y$ is a Banach space. Then $T$ has an extension\n$$\n\\tilde T:\\overline{\\mathcal{D}(T)}\\to Y.\n$$\nwhere $\\tilde T$ is a bounded linear operator with\n$$\n\\|\\tilde T\\|=\\|T\\|.\n$$\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"7-linear-functionals\"\u003e7. Linear Functionals\u003c/h2\u003e\n\u003ch3 id=\"71-definition-linear-functional\"\u003e7.1 Definition (Linear functional)\u003c/h3\u003e\n\u003cp\u003eA linear functional $f$ is a linear operator with domain in a vector space $X$ and range in the scalar field $K$ of $X$, thus\n$$\nf:\\mathcal{D}(f)\\to K.\n$$\nwhere $K=\\mathbb{R}$ if $X$ is real and $K=\\mathbb{C}$ if $X$ is complex.\u003c/p\u003e\n\u003ch3 id=\"72-definition-bounded-linear-functional\"\u003e7.2 Definition (Bounded linear functional)\u003c/h3\u003e\n\u003cp\u003eA bounded linear functional $f$ is a bounded linear operator with range in the scalar field of the normed space $X$ in which the domain $\\mathcal{D}(f)$ lies. Thus there exists a real number $c$ such that for all\n$x\\in\\mathcal{D}(f)$,\n$$\n|f(x)|\\le c\\|x\\|.\n$$\nFurthermore, the norm of $f$ is\n$$\n\\|f\\|=\\sup_{\\substack{x\\in\\mathcal{D}(f)\\\\ x\\ne 0}}\\frac{|f(x)|}{\\\n|x\\|}\n$$\nor\n$$\n\\|f\\|=\\sup_{\\substack{x\\in\\mathcal{D}(f)\\\\ \\|x\\|=1}}|f(x)|.\n$$\n$$\n\\Rightarrow |f(x)|\\le \\|f\\|\\|x\\|.\n$$\u003c/p\u003e\n\u003ch3 id=\"73-theorem-continuity-and-boundedness\"\u003e7.3 Theorem (Continuity and boundedness)\u003c/h3\u003e\n\u003cp\u003eA linear functional $f$ with domain $\\mathcal{D}(f)$ in a normed space is continuous if and only if $f$ is bounded.\u003c/p\u003e\n\u003ch3 id=\"examples\"\u003eExamples\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003e1. Norm.\u003c/strong\u003e The norm $\\|\\cdot\\|:X\\to\\mathbb{R}$ on a normed space $(X,\\|\\cdot\\|)$ is a functional on $X$ which is not linear.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2. Dot product.\u003c/strong\u003e The dot product with one factor kept fixed defines a functional\n$\nf:\\mathbb{R}^3\\to\\mathbb{R}\n$\n$$\nf(x)=x\\cdot a=\\xi_1 \\alpha_1+\\xi_2 \\alpha_2+\\xi_3 \\alpha_3.\n$$\nwhere $\\alpha=(\\alpha_j)\\in\\mathbb{R}^3$ is fixed.\u003cbr\u003e\n$f$ is linear, bounded. In fact,\n$$\n|f(x)|=|x\\cdot a|\\le \\|x\\|\\|a\\|\n\\Rightarrow \\|f\\|=\\|a\\|.\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3. Definite integral.\u003c/strong\u003e Consider integral for all functions in a certain function space, $f$ is defined by\n$$\nf(x)=\\int_a^b x(t)\\ \\mathrm{d}t, \\qquad x\\in C[a,b].\n$$\n$f$ is linear, bounded and has norm\n$\n\\|f\\|=b-a.\n$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e4. Space $C[a,b]$.\u003c/strong\u003e Choose a fixed $t_0\\in J=[a,b]$ and set\n$$\nf_t(x)=x(t_0), \\qquad x\\in C[a,b].\n$$\n$f_t$ is linear, $f_t$ is bounded and has norm\n$\n\\|f_t\\|=1.\n$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e5. Space $\\ell^2$.\u003c/strong\u003e We can obtain a linear functional $f$ on the Hilbert space $\\ell^2$ by choosing a fixed\n$\na=(a_j)\\in \\ell^2\n$\nand setting\n$$\nf(x)=\\sum_{j=1}^{\\infty}\\xi_j a_j,\n$$\nwhere\n$\nx=(\\xi_j)\\in \\ell^2.\n$\nThis series converges absolutely and $f$ is bounded.\u003c/p\u003e\n\u003cp\u003e\u0026ndash; The set of all linear functionals defined on a vector space $X$ can itself be made into a vector space. This space is denoted by $X^*$ and is called the \u003cstrong\u003ealgebraic dual space\u003c/strong\u003e of $X$.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eThe sum $f_1+f_2$ of two functionals $f_1$ and $f_2$ is the functional $s$ whose value at every $x\\in X$ is\n$$\ns(x)=(f_1+f_2)(x)=f_1(x)+f_2(x).\n$$\u003c/li\u003e\n\u003cli\u003eThe product of a scalar $\\alpha$ and a functional $f$ is the functional $p$ whose value at $x\\in X$ is\n$$\np(x)=(\\alpha f)(x)=\\alpha f(x).\n$$\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u0026ndash; Consider the algebraic dual $(X^{*})^{*}$ of $X^*$, whose elements are the linear functionals defined on $X^*$. Denote $(X^*)^*$ by $X^{* *}$ and call it the \u003cstrong\u003esecond algebraic dual space\u003c/strong\u003e of $X$.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eWe can obtain a $g\\in X^{**}$ which is a linear functional defined on $X^*$ by choosing a fixed $x\\in X$ and setting\n$$\ng(f)=g_x(f)=f(x) \\qquad (x\\in X \\text{ fixed}, f\\in X^* \\text{ variable}).\n$$\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u0026ndash; To each $x\\in X$ there corresponds a $g_x\\in X^{**}$. This defines a mapping\n\\begin{align}\nC:\\quad  \u0026amp; X \\to X^{**} \\\\\n\u0026amp; x \\text{ } \\mapsto g_x.\n\\end{align}\u003c/p\u003e\n\u003cp\u003e$C$ is called the \u003cstrong\u003ecanonical mapping\u003c/strong\u003e of $X$ into $X^{**}$.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eIsomorphism\u003c/strong\u003e. A bijective mapping of $X$ onto $\\tilde X$ which preserves the structure.\u003c/p\u003e\n\u003cp\u003eAccordingly, an isomorphism $T$ of a metric space $X=(X,d)$ onto a metric space $\\tilde X=(\\tilde X,\\tilde d)$ is a bijective mapping which preserves distance, that is, for all $x,y\\in X$,\n$$\n\\tilde d(Tx,Ty)=d(x,y).\n$$\u003c/p\u003e\n\u003cp\u003e$\\tilde X$ is then called \u003cem\u003eisomorphic\u003c/em\u003e with $X$. $T: X\\to \\tilde X$ is a bijective linear operator,\n$$\nT(x+y)=Tx+Ty, \\qquad T(\\alpha x)=\\alpha Tx.\n$$\u003c/p\u003e\n\u003cp\u003e$X$ and $\\tilde X$ are called isomorphic vector spaces.\u003c/p\u003e\n\u003cp\u003e\u0026ndash; Canonical mapping $C$ is injective. Since $C$ is linear, it is an isomorphism of $X$ onto the range $\\mathcal{R}(C)\\subset X^{**}$.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003eIf $X$ is isomorphic with a subspace of a vector space $Y$, we say that $X$ is \u003cem\u003eembeddable\u003c/em\u003e in $Y$. $X$ is embeddable in $X^{**}$ and $C$ is also called the \u003cem\u003ecanonical embedding\u003c/em\u003e of $X$ into $X^{**}$.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIf $C$ is surjective, so that $\\mathcal{R}(C)=X^{**}$, then $X$ is said to be \u003cem\u003ealgebraically reflexive\u003c/em\u003e.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"8-linear-operators-and-functionals-on-finite-dimensional-spaces\"\u003e8. Linear Operators and Functionals on Finite Dimensional Spaces\u003c/h2\u003e\n\u003cp\u003e\u0026ndash; Let $X$ and $Y$ be finite dimensional vector spaces over the same field and $T:X\\to Y$ a linear operator. We choose a basis\n$\nE=\\{e_1,\\dots,e_n\\} \\text{ for } X\n$\nand a basis\n$\nB=\\{b_1,\\dots,b_m\\} \\text{ for } Y,\n$\nwith the vectors arranged in a definite order which we keep fixed. Then every $x\\in X$ has a unique representation\n$$\nx=\\xi_1 e_1 + \\cdots + \\xi_n e_n.\n$$\u003c/p\u003e\n\u003cp\u003eSince $T$ is linear, $x$ has the image\n$$\ny = Tx = T\\left(\\sum_{k=1}^n \\xi_k e_k\\right) = \\sum_{k=1}^n \\xi_k T e_k.\n$$\u003c/p\u003e\n\u003cp\u003e$y$ and $y_k = T e_k$ are in $Y$; they have unique representations of the form\n$$\ny = \\sum_{j=1}^r \\eta_j b_j,\n$$\n$$\nT e_k = \\sum_{j=1}^r \\tau_{jk} b_j.\n$$\u003c/p\u003e\n\u003cp\u003eThen\n$$\ny = \\sum_{j=1}^r \\eta_j b_j\n= \\sum_{k=1}^n \\xi_k T e_k\n= \\sum_{k=1}^n \\xi_k \\sum_{j=1}^r \\tau_{jk} b_j\n= \\sum_{j=1}^r \\left(\\sum_{k=1}^n \\tau_{jk} \\xi_k \\right) b_j.\n$$\u003c/p\u003e\n\u003cp\u003e$$\n\\Rightarrow \\eta_j = \\sum_{k=1}^n \\tau_{jk} \\xi_k, \\qquad j=1,\\dots,r.\n$$\u003c/p\u003e\n\u003cp\u003eThe coefficients form a matrix:\n$\nT_{EB} = (\\tau_{jk})\n$\nwith $r$ rows and $n$ columns. The matrix $T_{EB}$ represents the operator $T$ with respect to bases.\u003c/p\u003e\n\u003cp\u003eLet $\\tilde{x} = (\\xi_k)$ and $\\tilde{y} = (\\eta_j)$. In matrix notation,\n$$\n\\tilde{y} = T_{EB}\\tilde{x},\n$$\n$$\nT e = T_{EB} b.\n$$\u003c/p\u003e\n\u003cp\u003e\u0026ndash; For every functional $f$ and every $x=\\sum\\limits_{j=1}^n \\xi_j e_j \\in X$, we have\n$$\nf(x)=f\\left(\\sum_{j=1}^n \\xi_j e_j\\right)\n= \\sum_{j=1}^n \\xi_j f(e_j)\n= \\sum_{j=1}^n \\xi_j \\alpha_j,\n$$\nwhere $\\alpha_j = f(e_j)$, and $f$ is uniquely determined by its values $\\alpha_j$ at the $n$ basis vectors of $X$. Conversely, every $n$-tuple of scalars $\\alpha_1,\\dots,\\alpha_n$ determines a linear functional on $X$.\nTake $n$-tuples\n$$\n(1,0,0,\\dots,0), \\quad (0,1,0,\\dots,0), \\quad \\dots, \\quad (0,0,0,\\dots,1).\n$$\u003c/p\u003e\n\u003cp\u003eThis gives $n$ functionals, denote $f_1,\\dots,f_n$, with values\n\u003cspan class=\"sidenote\" id=\"note1\"\u003e\n1. Kronecker delta\n\u003c/span\u003e\n$$\nf_j(e_k)=\\delta_{jk}^{\\hspace{0.2cm} \\color{gray}{1}}=\n\\begin{cases}\n0 \u0026amp; \\text{if } j\\ne k,\\\\\n1 \u0026amp; \\text{if } j=k,\n\\end{cases}\n$$\u003c/p\u003e\n\u003cp\u003e$\\{f_1,\\dots,f_n\\}$ is called the \u003cstrong\u003edual basis\u003c/strong\u003e of the basis $\\{e_1,\\dots,e_n\\}$ of $X$.\u003c/p\u003e\n\u003ch3 id=\"81-theorem-dimension-of-x\"\u003e8.1 Theorem (Dimension of $X^*$)\u003c/h3\u003e\n\u003cp\u003eLet $X$ be an $n$-dimensional vector space and $E=\\{e_1,\\dots,e_n\\}$ a basis for $X$. Then\n$\nF=\\{f_1,\\dots,f_n\\}\n$\nis a basis for the algebraic dual $X^*$ of $X$, and\n$\n\\dim X^* = \\dim X = n.\n$\u003c/p\u003e\n\u003ch3 id=\"82-lemma-zero-vector\"\u003e8.2 Lemma (Zero vector)\u003c/h3\u003e\n\u003cp\u003eLet $X$ be a finite dimensional vector space. If $x_0 \\in X$ has the property that\n$\nf(x_0)=0\n$\nfor all $f\\in X^*$, then $x_0=0$.\u003c/p\u003e\n\u003ch3 id=\"83-theorem-algebraic-reflexivity\"\u003e8.3 Theorem (Algebraic reflexivity)\u003c/h3\u003e\n\u003cp\u003eA finite dimensional vector space is algebraically reflexive.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"9-normed-spaces-of-operators-dual-space\"\u003e9. Normed Spaces of Operators. Dual Space.\u003c/h2\u003e\n\u003cp\u003e\u0026ndash; Take any two normed spaces $X$ and $Y$ (both real or complex) and consider the set $B(X,Y)$ consisting of all bounded linear operators from $X$ onto $Y$. $B(X,Y)$ can itself be made into a normed space.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003eDefine $T_1+T_2$ of two operators $T_1,T_2\\in B(X,Y)$ by\n$$\n(T_1+T_2)x = T_1x + T_2x.\n$$\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eDefine the product $\\alpha T$ of $T\\in B(X,Y)$ and a scalar $\\alpha$ by\n$$\n(\\alpha T)x = \\alpha Tx.\n$$\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThen $B(X,Y)$ becomes a vector space so itself a normed space with norm defined by\n$$\n\\|T\\| = \\sup_{\\substack{x\\in X \\\\ x\\ne 0}} \\frac{\\|Tx\\|}{\\|x\\|}\n= \\sup_{\\|x\\|=1} \\|Tx\\|.\n$$\u003c/p\u003e\n\u003ch3 id=\"91-theorem-completeness\"\u003e9.1 Theorem (Completeness)\u003c/h3\u003e\n\u003cp\u003eIf $Y$ is a Banach space, then $B(X,Y)$ is a Banach space.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eConsider an arbitrary Cauchy sequence $(T_n)$ in $B(X,Y)$ and show that $(T_n)$ converges to an operator $T\\in B(X,Y)$.\u003c/p\u003e\n\u003cp\u003eSince $(T_n)$ is Cauchy, for every $\\varepsilon\u0026gt;0$ there is an $N$ such that\n$$\n\\|T_n - T_m\\| \u0026lt; \\varepsilon \\quad (m,n \u0026gt; N).\n$$\u003c/p\u003e\n\u003cp\u003eFor all $x\\in X$ and $m,n\u0026gt;N$ we obtain\n\\begin{equation}\n\\|T_n x - T_m x\\| = \\|(T_n - T_m)x\\| \\le \\|T_n - T_m\\|\\|x\\| \u0026lt; \\varepsilon \\|x\\| \\tag{1}.\n\\end{equation}\u003c/p\u003e\n\u003cp\u003e$\\Rightarrow$ for fixed $x$, $(T_n x)$ is Cauchy in $Y$. Since $Y$ is complete, $(T_n x)$ converges, say\n$$\nT_n x \\to y.\n$$\nThe limit $y\\in Y$ depends on the choice of $x\\in X$. Define an operator\n$$\nT: X \\to Y, \\quad \\text{where } y=Tx.\n$$\u003c/p\u003e\n\u003cp\u003eThe operator $T$ is linear:\n$$\n\\lim T_n(\\alpha x_1 + \\beta x_2)\n= \\lim (\\alpha T_n x_1 + \\beta T_n x_2)\n= \\alpha \\lim T_n x_1 + \\beta \\lim T_n x_2.\n$$\u003c/p\u003e\n\u003cp\u003eWe prove that $T$ is bounded and $T_n \\to T$, that is, $\\|T_n - T\\|\\to 0$. Since (1) holds for every $m\u0026gt;N$ and $T_m x \\to Tx$, let $m\\to\\infty$. Using the continuity of the norm, for every $n\u0026gt;N$ and all $x\\in X$,\n\\begin{equation}\n\\|T_n x - Tx\\|\n= \\|T_n x - \\lim_{m\\to\\infty} T_m x\\|\n= \\lim_{m\\to\\infty} \\|T_n x - T_m x\\|\n\\le \\varepsilon \\|x\\|. \\tag{2}\n\\end{equation}\u003c/p\u003e\n\u003cp\u003e$\\Rightarrow (T_n - T)$ with $n\u0026gt;N$ is a bounded linear operator. Since $T_n$ is bounded, $T = T_n - (T_n - T)$ is bounded, hence $T\\in B(X,Y)$. In (2), take the supremum over all $x$ of norm $1$, we obtain\n$$\n\\|T_n - T\\| \\le \\varepsilon \\quad (n\u0026gt;N).\n$$\u003c/p\u003e\n\u003cp\u003eHence, $\\|T_n - T\\|\\to 0$.\u003c/p\u003e\n\u003ch3 id=\"92-definition-dual-space-x\"\u003e9.2 Definition (Dual space $X\u0026rsquo;$)\u003c/h3\u003e\n\u003cp\u003eLet $X$ be a normed space. Then the set of all bounded linear functionals on $X$ constitutes a normed space with norm defined by\u003c/p\u003e\n\u003cp\u003e$$\n\\|f\\|=\\sup_{\\substack{x\\in X \\\\ x\\neq 0}}\\frac{|f(x)|}{\\|x\\|}=\\sup_{\\|x\\|=1}|f(x)|.\n$$\u003c/p\u003e\n\u003cp\u003ewhich is called the dual space of $X$ and is denoted by $X\u0026rsquo;$.\u003c/p\u003e\n\u003ch3 id=\"93-theorem-dual-space\"\u003e9.3 Theorem (Dual Space)\u003c/h3\u003e\n\u003cp\u003eThe dual space $X\u0026rsquo;$ of a normed space $X$ is a Banach space.\u003c/p\u003e\n\u003cp\u003eAn isomorphism of a normed space $X$ onto a normed space $\\tilde X$ is a bijective linear operator $T:X\\to \\tilde X$ which preserves the norm, that is, for all $x\\in X$,\n$\n\\|Tx\\|=\\|x\\|.\n$\n($T$ is isometric)\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e$X$ is called isomorphic with $\\tilde X$.\u003c/li\u003e\n\u003cli\u003e$X$ and $\\tilde X$ are called isomorphic normed space.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"example-4\"\u003eExample\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003e1. Space $\\mathbb{R}^n$.\u003c/strong\u003e The dual space of $\\mathbb{R}^n$ is $\\mathbb{R}^n$.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eWe have $(\\mathbb{R}^{n})^{'}=(\\mathbb{R}^{n})^*$ and every $f\\in (\\mathbb{R}^{n})^*$ has a representation\u003c/p\u003e\n\u003cp\u003e$$\nf(x)=\\sum \\xi_k \\gamma_k,\\qquad \\gamma_k=f(e_k).\n$$\u003c/p\u003e\n\u003cp\u003eBy Cauchy\u0026ndash;Schwarz inequality,\n$$\n|f(x)|\\le \\sum |\\xi_k \\gamma_k|\n\\le \\left(\\sum \\xi_j^2\\right)^{1/2}\\left(\\sum \\gamma_k^2\\right)^{1/2}\n=\\|x\\|\\left(\\sum \\gamma_k^2\\right)^{1/2}.\n$$\nTaking the supremum over all $x$ of norm $1$, we obtain\n$$\n\\|f\\|\\le \\left(\\sum \\gamma_k^2\\right)^{1/2}.\n$$\nHowever, since for $x=(x_1,\\ldots,x_n)\\to$ equality is achieved, we must have\u003c/p\u003e\n\u003cp\u003e$$\n\\|f\\|=\\left(\\sum_{k=1}^n \\gamma_k^2\\right)^{1/2}.\n$$\u003c/p\u003e\n\u003cp\u003eThis proves that norm of $f$ is the Euclidean norm, $\\|f\\|=\\|c\\|$ where $c=(\\gamma_k)\\in \\mathbb{R}^n$.\u003c/p\u003e\n\u003cp\u003eHence the mapping of $(\\mathbb{R}^{n})^{\u0026rsquo;}$ onto $\\mathbb{R}^n$ defined by $f\\mapsto c=(\\gamma_k)$, $\\gamma_k=f(e_k)$, is norm preserving, it is linear and bijective, it is an isomorphism.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2. Space $\\ell^1$.\u003c/strong\u003e The dual space of $\\ell^1$ is $\\ell^\\infty$.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3. Space $\\ell^p$.\u003c/strong\u003e The dual space of $\\ell^p$ is $\\ell^q$; here, $1 \u0026lt; p \u0026lt; +\\infty$ and $q$ is the conjugate of $p$, that is, $1/p + 1/q = 1.$\u003c/p\u003e\n\u003ch2 id=\"references\"\u003eReferences\u003c/h2\u003e\n\u003cp\u003e[1] Kreyszig, E. (1978). \u003cem\u003eIntroductory functional analysis with applications\u003c/em\u003e. John Wiley \u0026amp; Sons.\u003c/p\u003e\n  \u003c/div\u003e\n\u003cp\u003e\n  \u003ca href=\"mailto:nguyencungthanh11@gmail.com?subject=Re: Banach%20space\" rel='noopener noreferrer'\u003e💬 Reply via e-mail\u003c/a\u003e\n\u003c/p\u003e\n","date_modified":"2025-10-08T00:00:00+00:00","date_published":"2025-10-08T00:00:00+00:00","id":"https://nguyencungthanh.github.io/2025/10/08/banach-space/","summary":"","tags":["Functional Analysis","Banach Space"],"title":"Banach space","url":"https://nguyencungthanh.github.io/2025/10/08/banach-space/"},{"content_html":"\n  \u003cdiv class='e-content'\u003e\n    \u003cscript\u003e\nwindow.MathJax = {\n    loader: {load: ['[tex]/ams']},\n    tex: {\n        inlineMath: [['$','$'], ['\\\\(','\\\\)']],\n        packages: {'[+]': ['ams']}\n    },\n    svg: {fontCache: 'global'},\n    useLabelIds: true\n};\n\u003c/script\u003e\n\u003cscript async src=\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-svg.js\"\u003e\u003c/script\u003e\n\u003cstyle\u003e\nmjx-container[jax=\"SVG\"][display=\"true\"] {\n  display: block;\n  text-align: center;\n  margin: 1em auto;\n}\n\u003c/style\u003e\n\u003ch2 id=\"1-definition\"\u003e1. Definition\u003c/h2\u003e\n\u003cp\u003e\u0026ndash; A metric space is a pair $(X, d)$, where $X$ is a set and $d$ is a metric on $X$ (or distance function on $X$), that is a function defined on $X \\times X$ such that for all $x, y, z \\in X$, we have\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e$d$ is real - valued, finite and non negative\u003c/li\u003e\n\u003cli\u003e$d(x, y)=0 \\Leftrightarrow x=y$\u003c/li\u003e\n\u003cli\u003e$d(x, y)=d(y, x)$\u003c/li\u003e\n\u003cli\u003e$d(x, y) \\leq d(x, z)+d(z, y) \\rightarrow d\\left(x, x_{n}\\right) \\leq \\sum_{i=1}^{n-1} d\\left(x_{i}, x_{i+1}\\right)$\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u0026ndash; A subspace $(Y, \\tilde{d})$ of $(X, d)$ is obtained if we take a subset $Y_{C} \\subset X$ and restrict $d$ to $Y \\times Y$; thus the metric on $Y$ is the restriction\n$$\n\\tilde{d}= d|_{Y \\times Y}\n$$\n$\\tilde{d}$ is called the metric induced on $y$ by $d$\u003c/p\u003e\n\u003ch3 id=\"example\"\u003eExample\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003e1. Euclidean space $\\mathbb{R}^{n}$, unitary space $\\mathbb{C}^{n}$, complex plan $\\mathbb{C}.$\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eTake the set of all ordered $n$-tuples of real numbers, written\n$$\nx=\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right), \\quad y=\\left(y_{1}, y_{2}, \\ldots, y_{n}\\right)\n$$\nand the Euclidean metric defined by\n$$\nd(x, y)=\\sqrt{\\left(x_{1}-y_{1}\\right)^{2}+\\left(x_{2}-y_{2}\\right)^{2}+\\ldots+\\left(x_{n}-y_{n}\\right)^{2}}\n$$\n$n$ - dimensional unitary space $\\mathbb{C}^{n}$ has the metric defined by\n$$\nd(x, y)=\\sqrt{\\left|x_{1}-y_{1}\\right|^{2}+\\left|x_{2}-y_{2}\\right|^{2}+\\ldots+\\left|x_{n}-y_{n}\\right|^{2}}\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2. Sequence space $l^{\\infty}$.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eTake the set of all bounded sequences of complex numbers; every element of $X$ is a complex sequence\n$$\nx=\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right) \\text { briefly } x=\\left(x_{i}\\right)\n$$\nsuch that for all $j \\in \\mathbb{N}^{*}$, we have\u003c/p\u003e\n\u003cp\u003e$$\n\\left|x_{j}\\right| \\leqslant c_{x}\n$$\n$c_{x}$ is a real number which may depend on $x$, not on $j$. Metric is defined by\n$$\nd(x, y)=\\sup_{j \\in \\mathbb{N}^{*}}|x_{j}-y_{j}|\n$$\nwhere $y=\\left(y_{j}\\right) \\in X$.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3. Function space $C[a, b]$.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eTake the set of all real-valued function $x, y, \\ldots$ and are defined and continuous on a given closed interval $J=[a, b]$ The metric defined by\n$$\nd(x, y)=\\max _{t \\in J}|x(t)-y(t)|\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e4. Sequence spaces.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe space consists of the set of all (bounded or unbounded) sequences of complex numbers and the metric $d$ defined by\nwhere $x=\\left(x_{j}\\right), y=\\left(y_{j}\\right)$\n$$\nd(x, y)=\\sum_{j=1}^{\\infty} \\frac{1}{2^{j}} \\frac{\\left|x_{j}-y_{j}\\right|}{1+\\left|x_{j}-y_{j}\\right|}\n$$\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eProof.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e$f(t)=\\frac{t}{1+t}$ is monotone increasing; $|a+b| \\leqslant|a|+|b|$ implies $f(|a+b|) \\leqslant f(|a|+|b|).$\nThen we have:\n$$\n\\frac{|a+b|}{1+|a+b|} \\leqslant \\frac{|a|+|b|}{1+|a|+|b|} \\leqslant \\frac{|a|}{1+|a|}+\\frac{|b|}{1+|b|}\n$$\u003c/p\u003e\n\u003cp\u003eLet $a=x_{j}-z_{j}$ and $y=z_{j}-y_{j}$ where $z=\\left(z_{j}\\right)$. Therefore $a+b=x_{j}-y_{j}$ and\n$$\n\\frac{\\left|x_{j}-y_{j}\\right|}{1+\\left|x_{j}-y_{j}\\right|} \\leqslant \\frac{\\left|x_{j}-z_{j}\\right|}{1+\\left|x_{j}-z_{j}\\right|}+\\frac{\\left|z_{j}-y_{j}\\right|}{1+\\left|z_{j}-y_{j}\\right|}\n$$\nThen\n$$ \\sum_{j=1}^{\\infty} \\frac{1}{2^{j}} \\frac{\\left|x_{j}-y_{j}\\right|}{1+\\left|x_{j}-y_{j}\\right|} \\leqslant \\sum_{j=1}^{\\infty} \\frac{1}{2^{j}} \\frac{\\left|x_{j}-z_{j}\\right|}{1+\\left|x_{j}-z_{j}\\right|}+\\sum_{j=1}^{\\infty} \\frac{1}{2^{j}} \\frac{\\left|z_{j}-y_{j}\\right|}{1+\\left|z_{j}-y_{j}\\right|}$$\nor $$d(x, y) \\leq d(x, z)+d(z, y).$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e5. Space $B(A)$ of bounded function.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eEach element $x \\in B(A)$ is a function defined and bounded on a given set $A$, the metric is defined:\n$$\nd(x, y)=\\sup _{t \\in A}|x(t)-y(t)|\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e6. Space $l^{p}$, Hillbert sequence space $l^{2}$, Holder and Minkowski inequalities for sums.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eEach element in the space $\\ell^{p}$ is a sequence $x=\\left(x_{i}\\right)=\\left(x_{1}, x_{2}, \\ldots\\right)$ of numbers such that\n$$\n\\sum_{j=1}^{\\infty}\\left|x_{i}\\right|^{p}\u0026lt;\\infty \\quad(p \\geqslant 1) \\\n$$\nand the metric is defined by\n$$\nd(x, y)=\\left(\\sum_{j=1}^{\\infty}\\left|x_{j}-y_{j}\\right|^{p}\\right)^{\\frac{1}{p}}\n$$\n\u003cem\u003eProof.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e$l^{p}$ is a metric space\u003c/p\u003e\n\u003cp\u003e$\\text{(a)} \\quad$ Auxiliary in equality\n$\\frac{1}{p}+\\frac{1}{q}=1$ ($p, q$ are called conjugate exponents). Then\n$$\n\\alpha \\beta \\leqslant \\frac{\\alpha^{p}}{p}+\\frac{\\beta^{q}}{q}\n$$\u003c/p\u003e\n\u003cp\u003e$\\text{(b)} \\quad$ Holder inequality\n$$\n\\sum_{j=1}^{\\infty}\\left|x_{j} y_{j}\\right| \\leqslant\\left(\\sum_{j=1}^{\\infty}\\left|x_{j}\\right|^{p}\\right)^{\\frac{1}{p}}\\left(\\sum_{j=1}^{\\infty}\\left|y_{j}\\right|^{q}\\right)^{\\frac{1}{q}}\n$$\u003c/p\u003e\n\u003cp\u003eLet $ \\left(\\tilde{x_{i}}\\right)$ and $\\left(\\tilde{y_{j}}\\right) $ such that\u003c/p\u003e\n\u003cp\u003e$$\n\\begin{equation}\n\\sum\\left|\\tilde{x_{i}}\\right|^{p} = 1, \\quad \\sum\\left|\\tilde{y_{j}}\\right|^{q}=1 \\tag{1}\n\\end{equation}\n$$\u003c/p\u003e\n\u003cp\u003eSet $\\alpha=\\left|x_{j}\\right|$ and $\\beta=\\left|y_{j}\\right|$, we have\n$$\n\\begin{equation}\n\\left|\\tilde{x_{j}} \\tilde{y_{j}}\\right| \\leqslant \\frac{1}{p}\\left|\\tilde{x_{j}}\\right|^{p}+\\frac{1}{q}\\left|\\tilde{y_{j}}\\right|^{q} \\tag{2}\n\\end{equation}\n$$\u003c/p\u003e\n\u003cp\u003eThen\n$$\n\\begin{equation}\n\\sum\\left|\\tilde{x_{j}}\\tilde{y_{j}}\\right| \\leqslant \\frac{1}{p}+\\frac{1}{q}=1. \\tag{3}\n\\end{equation}\n$$\u003c/p\u003e\n\u003cp\u003eNow, take $x=\\left(x_{j}\\right) \\in l^{p}$ and $y=\\left(y_{j}\\right) \\in l^{p}$ and set\n$$\n\\tilde{x_{j}}=\\dfrac{x_{j}}{\\left(\\sum\\left|x_{k}\\right|^{p}\\right)^{\\frac{1}{p}}}; \\quad \\tilde{y_{j}}=\\dfrac{y_{j}}{\\left(\\sum\\left|y_{j}\\right|^{q}\\right)^{\\frac{1}{q}}}\n$$\u003c/p\u003e\n\u003cp\u003eThen (1) sastified, so we apply (2) and (3) to have\n$$\n\\sum_{j=1}^{\\infty}\\left|x_{j} y_{j}\\right| \\leqslant\\left(\\sum_{k=1}^{\\infty}\\left|x_{k}\\right|^{p}\\right)^{\\frac{1}{p}}\\left(\\sum_{m=1}^{\\infty}\\left|y_{m}\\right|^{q}\\right)^{\\frac{1}{q}}\n$$\u003c/p\u003e\n\u003cp\u003e$\\text{(c)} \\quad$ Minkowski inequality\n$$\n\\begin{equation}\n\\left(\\sum_{j=1}^{\\infty}\\left|x_{j}+y_{j}\\right|^{p}\\right)^{\\frac{1}{p}} \\leqslant\\left(\\sum_{k=1}^{\\infty}\\left|x_{k}\\right|^{p}\\right)^{\\frac{1}{p}}+\\left(\\sum_{m=1}^{\\infty}\\left|y_{m}\\right|^{p}\\right)^{\\frac{1}{p}} \\tag{4}\n\\end{equation}\n$$\nwhere $x=\\left(x_{j}\\right) \\in l^{p}$ and $y=\\left(y_{j}\\right) \\in l^{p}$.\u003c/p\u003e\n\u003cp\u003eWrite $x_{j}+y_{j}=\\omega_{j}$, then\n$$\n\\begin{aligned}\n\\left|w_{j}\\right|^{p} \u0026amp; =\\left|x_{j}+y_{j}\\right|\\left|w_{j}\\right|^{p-1} \\\\\n\u0026amp; \\leqslant \\left( \\left|x_{j}\\right|+\\left|y_{j}\\right| \\right) \\left|w_{j}\\right|^{p-1}\n\\end{aligned}\n$$\u003c/p\u003e\n\u003cp\u003eSumming over $j$ from 1 to any fixed $n$, we obtain\n$$\n\\begin{align}\n\\sum\\left|w_{j}\\right|^{p} \u0026amp;\\leqslant \\sum\\left|x_{j}\\right|\\left|w_{j}\\right|^{p-1}+\\sum\\left|y_{j}\\right|\\left|w_{j}\\right|^{p-1} \\\\\n\u0026amp;\\leqslant \\left(\\sum\\left|x_{k}\\right|^{p}\\right)^{\\frac{1}{p}}\\left(\\sum\\left|w_{m}\\right|^{(p-1)q}\\right)^{\\frac{1}{q}}+\\left(\\sum\\left|w_{k}\\right|^{p}\\right)^{\\frac{1}{p}}\\left(\\sum\\left|w_{m}\\right|^{(p-1) q}\\right)^{\\frac{1}{q}} \\\\\n\u0026amp;= \\left\\{\\left(\\sum\\left|x_{k}\\right|^{p}\\right)^{\\frac{1}{p}}+\\left(\\sum\\left|y_{k}\\right|^{p}\\right)^{\\frac{1}{p}}\\right\\}\\left(\\sum\\left|w_{m}\\right|^{p}\\right)^{\\frac{1}{q}} \\quad ((p-1) q=p)\n\\end{align}\n$$\u003c/p\u003e\n\u003cp\u003eDividing by the last factor and noting that $1-\\frac{1}{q}=\\frac{1}{p}$, we obtain (4).\u003c/p\u003e\n\u003cp\u003e$\\text{(d)} \\quad$ From $\\text{(c)}$\n$$\n\\begin{align}\nd(x, y) \u0026amp; =\\left(\\sum\\left|x_{j}-y_{j}\\right|^{p}\\right)^{\\frac{1}{p}} \\\\\n\u0026amp; \\leqslant\\left(\\sum\\left(\\left|x_{j}-z_{j}\\right|+\\left|z_{j}-y_{j}\\right|^{p}\\right)^{\\frac{1}{p}}\\right. \\\\\n\u0026amp; \\leqslant\\left(\\sum\\left|x_{j}-z_{j}\\right|^{p}\\right)^{\\frac{1}{p}}+\\left(\\sum\\left|z_{j}-y_{j}\\right|^{p}\\right)^{\\frac{1}{p}} \\\\\n\u0026amp; =d(x, z)+d(z, y)\n\\end{align}\n$$\u003c/p\u003e\n\u003ch3 id=\"diameter-bounded-set\"\u003eDiameter, bounded set\u003c/h3\u003e\n\u003cp\u003eThe diameter $\\delta(A)$ of a nonempty set $A$ in a metric space $(X, d)$ is defined to be\n$$\n\\delta(A)=\\sup _{x, y \\in A} d(x, y) .\n$$\n$A$ is said to be bounded if $\\delta(A)\u0026lt;\\infty$\u003c/p\u003e\n\u003ch3 id=\"distance-between-sets\"\u003eDistance between sets\u003c/h3\u003e\n\u003cp\u003eThe distance $D(A, B)$ between two nonempty subsets $A$ and $B$ of a metric space $(x, d)$ is defined to be\n$$\nD(A, B)=\\inf _{\\substack{a \\in A, \\\\ b \\in B}} d(a, b) .\n$$\n$D$ does not define a metric on the power set of $X$\u003c/p\u003e\n\u003ch3 id=\"product-of-metric-space\"\u003eProduct of metric space\u003c/h3\u003e\n\u003cp\u003eThe Cartesian product $X=X_{1} \\times X_{2}$ of two metric spaces $\\left(X_{1}, d_{1}\\right)$ and $\\left(X_{2}, d_{2}\\right)$ can be made into a metric space $(X, d)$.\u003c/p\u003e\n\u003ch3 id=\"example-1\"\u003eExample\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003e1.\u003c/strong\u003e $d(x, y)=d_{1}\\left(x_{1}, y_{1}\\right)+d_{2}\\left(x_{2}, y_{2}\\right)$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2.\u003c/strong\u003e $\\tilde{d}(x, y)=\\sqrt{d_{1}\\left(x_{1}, y_{1}\\right)^{2}+d_{2}\\left(x_{2}, y_{2}\\right)^{2}}$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.\u003c/strong\u003e $\\tilde{d}(x, y)=\\max \\left[d_{1}\\left(x_{1}, y_{1}\\right), d_{2}\\left(x_{2}, y_{2}\\right)\\right]$\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"2-open-set-closed-set-neighborhood\"\u003e2. Open Set, Closed Set, Neighborhood\u003c/h2\u003e\n\u003ch3 id=\"21-definition-ball-and-sphere\"\u003e2.1 Definition (Ball and sphere)\u003c/h3\u003e\n\u003cp\u003eGiven a point $x_{0} \\in X$ and a real number $r\u0026gt;0$, we define three types of sets\u003c/p\u003e\n\u003cp\u003e$\\text{(a)} \\quad$ $B\\left(x_{0}, r\\right)=\\left\\{x \\in X \\mid d\\left(x, x_{0}\\right)\u0026lt;r\\right\\} \\quad$ (Open ball)\u003c/p\u003e\n\u003cp\u003e$\\text{(b)} \\quad$ $\\tilde{B}\\left(x_{0}, r\\right)=\\left\\{x \\in X \\mid d\\left(x, x_{0}\\right) \\leq r\\right\\}\n\\quad$ (Closed ball)\u003c/p\u003e\n\u003cp\u003e$\\text{(c)} \\quad$ $S\\left(x_{0}, r\\right)=\\left\\{x \\in X \\mid d\\left(x, x_{0}\\right)=r\\right\\} \\quad$ (Sphere)\u003c/p\u003e\n\u003cp\u003e$x_{0}$ : center; $r$ : radius.\n$S\\left(x_{0}, r\\right)=\\tilde{B}\\left(x_{0}, r\\right)-B\\left(x_{0}, r\\right)$.\u003c/p\u003e\n\u003ch3 id=\"22-definition-open-set-closed-set\"\u003e2.2 Definition (Open set, Closed set).\u003c/h3\u003e\n\u003cp\u003eA subset $M$ of a metric space $X$ is said to be open if it contains a ball about each of its points. $A$ subset $K$ of $X$ is said to be closed if its complement (in $X$ ) is open, that is, $K^{c}=X-K$ is open.\u003c/p\u003e\n\u003cp\u003e\u0026ndash; An open ball $B\\left(x_{0}, \\varepsilon\\right)$ of radius $\\varepsilon$ is often called an $\\varepsilon$-neighborhood of $x_{0}$\u003c/p\u003e\n\u003cp\u003e\u0026ndash; Call $x_{0}$ an interior point of a set $M \\subset X$ if $M$ is a neighborhood of $x_{0}$.\u003c/p\u003e\n\u003cp\u003e\u0026ndash; The interior of $M$ is the set of all interior points of $M$, denoted $M^{0}$ or $\\operatorname{Int}(M)$.\u003c/p\u003e\n\u003cp\u003e\u0026ndash; The collection of all open subsets of $X$, call it $\\mathcal{J}$, has properties:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003e$T_{1}. \\emptyset \\in \\mathcal{J}, x \\in \\mathcal{J}$\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e$T_{2}$. The union of any nembers of $\\mathcal{J}$ is a member of $\\mathcal{J}$\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e$T_{3}$. The intersection of finitely many members of $\\mathcal{J}$ is a member of $\\mathcal{J}$.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u0026ndash; Define a topological space $(x, g)$ to be a set $x$ and a collection $\\mathcal{J}$ of subsets of $x$ such that $\\mathcal{J}$ satisfies $\\left(T_{1}\\right)$ and $\\left(T_{3}\\right)$\u003c/p\u003e\n\u003cp\u003e\u0026ndash; $\\mathcal{J}:$ topology for $X$.\u003c/p\u003e\n\u003ch3 id=\"23-definition-continuous-mapping\"\u003e2.3 Definition (Continuous mapping).\u003c/h3\u003e\n\u003cp\u003eLet $X=(x, d)$ and $Y=(y, \\tilde{d})$ be metric spaces. A mapping $T: X \\rightarrow Y$ is said to be continuous at a point $x_{0} \\in X$ if for every $\\varepsilon\u0026gt;0$, there is a $\\delta\u0026gt;0$ such that\n$$\n\\tilde{d}\\left(T x, T x_{0}\\right)\u0026lt;\\varepsilon \\quad \\text { for all } x \\text { satisfying } \\quad d\\left(x, x_{0}\\right)\u0026lt;\\delta\n$$\u003c/p\u003e\n\u003ch3 id=\"24-theorem-continuous-mapping\"\u003e2.4 Theorem (Continuous mapping)\u003c/h3\u003e\n\u003cp\u003eA mapping $T$ of a metric space $X$ into a metric space $Y$ is continuous if and only if the inverse image of any open subset of $Y$ is an open subset of $X$.\u003c/p\u003e\n\u003cp\u003eLet $M$ be a subset space $X$. Then a point $x_{0}$ of $X$ is called an accumulation point of $M$ if every neighborhood of $x_{0}$ contains at least one point $y \\in M$ distance from $x_{0}$. The set consisting of the points of $M$ and the accumulation points of $M$ is called the closure of $M$ and is denoted by $\\bar{M}$.\u003c/p\u003e\n\u003ch3 id=\"25-definition-dense-set-separable-space\"\u003e2.5 Definition (Dense set, separable space)\u003c/h3\u003e\n\u003cp\u003eA subset $M$ of a metric space $X$ is said to be dense in $X$ if $\\bar{M}=X$.\n$X$ is said to be separable if it has countable subset which is dense in $X$\u003c/p\u003e\n\u003ch3 id=\"example-2\"\u003eExample\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003e1. Real line $\\mathbb{R}$.\u003c/strong\u003e The real line $\\mathbb{R}$ is separable.\n\u0026ndash; The set $\\mathbb{Q}$ of all rational numbers is countable and is dense in $\\mathbb{R}$.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2. Complex plane $\\mathbb{C}$.\u003c/strong\u003e The complex plane $\\mathbb{C}$ is separable\n\u0026ndash; A countable dense subset of $\\mathbb{C}$ is the set of all complex numbers whose real and imaginary parts are both rational.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3 Discrete metric space.\u003c/strong\u003e A discre metric space $X$ is separable if and only if $X$ is countable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e4. Space $l^{\\infty}$.\u003c/strong\u003e The space $l^{\\infty}$ is not separable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e5. Space $\\ell^{p}$.\u003c/strong\u003e The space $\\ell^{p}$ with $1 \\leqslant p\u0026lt;\\infty$ is separable\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"3-convergence-cauchy-sequence-completeness\"\u003e3. Convergence, Cauchy Sequence, Completeness.\u003c/h2\u003e\n\u003ch3 id=\"31-definition-convergence-of-a-sequence-limit\"\u003e3.1 Definition (Convergence of a sequence, limit)\u003c/h3\u003e\n\u003cp\u003eA sequence $\\left(x_{n}\\right)$ in a metric space $X=(x, d)$ is said to converge or to be convergent if there is an $x \\in X$ such that\n$$\n\\lim_{n \\rightarrow \\infty} d\\left(x_{n}, x\\right)=0\n$$\n$x$ is called the limit of $\\left(x_{n}\\right)$ and we write\n$$\n\\lim_{n \\rightarrow \\infty} x_{n}=x\n$$\nor, simply,\n$$\n\\lim_{n \\rightarrow \\infty} x_{n} \\to x\n$$\nHence, if $x_{n} \\rightarrow x$, an $\\varepsilon\u0026gt;0$ being given, there is an $N=N(\\varepsilon)$ such that all $x_{n}$ with $n\u0026gt;N$ lie in the $\\varepsilon$ - neighborhood $B(x, \\varepsilon)$ of $x$.\u003c/p\u003e\n\u003ch3 id=\"32-lemma-boundedness-limit\"\u003e3.2 Lemma (Boundedness limit)\u003c/h3\u003e\n\u003cp\u003eLet $(x, d)$ be a metric space. Then:\u003c/p\u003e\n\u003cp\u003e$\\text{(a)} \\quad$ A convergent sequence in $x$ is bounded and its limit is unique.\u003c/p\u003e\n\u003cp\u003e$\\text{(b)} \\quad$ If $x_{n} \\rightarrow x$ and $y_{n} \\rightarrow y$ in $x$, then $d\\left(x_{n}, y_{n}\\right) \\rightarrow d(x, y)$.\u003c/p\u003e\n\u003ch3 id=\"33-definition-cauchy-sequence-completeness\"\u003e3.3 Definition (Cauchy sequence, completeness)\u003c/h3\u003e\n\u003cp\u003eA sequence $\\left(x_{n}\\right)$ in a metric space $X=(x, d)$ is said to be cauchy if for every $\\varepsilon\u0026gt;0$, there is a $N=N(\\varepsilon)$ such that\n$$\nd\\left(x_{m}, x_{n}\\right)\u0026lt;\\varepsilon \\quad \\forall m, n\u0026gt;N .\n$$\u003c/p\u003e\n\u003cp\u003eThe space $X$ is said to be complete if every Cauchy sequence in $X$ converges (has a limit which is an element of $x$).\u003c/p\u003e\n\u003ch3 id=\"34-theorem-convergent-sequence\"\u003e3.4 Theorem (Convergent sequence)\u003c/h3\u003e\n\u003cp\u003eEvery convergent sequence in a metric space is a Cauchy sequence.\u003c/p\u003e\n\u003ch3 id=\"35-theorem-closure-closed-set\"\u003e3.5 Theorem (Closure, closed set)\u003c/h3\u003e\n\u003cp\u003eLet $M$ be a nonempty subset of metric space ($X, d$) and $\\bar{M}$ its closure.\u003c/p\u003e\n\u003cp\u003e$\\text{(a)} \\quad$ $x \\in \\bar{M}$ if and only if there is a sequence $\\left(x_{n}\\right)$ in $M$ such that $x_{n} \\rightarrow x$.\u003c/p\u003e\n\u003cp\u003e$\\text{(b)} \\quad$ $M$ is closed if and only if the situation $x_{n} \\in M, x_{n} \\rightarrow x$ implies that $x \\in M$.\u003c/p\u003e\n\u003ch3 id=\"36-theorem-complete-space\"\u003e3.6 Theorem (Complete space)\u003c/h3\u003e\n\u003cp\u003eA subspace $M$ of a complete metric space $X$ is itself complete if and only if the set $M$ is closed in $X$.\u003c/p\u003e\n\u003ch3 id=\"37-theorem-continuous-mapping\"\u003e3.7 Theorem (Continuous mapping)\u003c/h3\u003e\n\u003cp\u003eA mapping $T: X \\rightarrow Y$ of a metric space $(X, d)$ into a metric space $(Y, \\tilde{d})$ is continuous at a point $x_{0} \\in X$ if and only if $x_{n} \\rightarrow x_{0}$ implies $Tx_{n} \\rightarrow Tx_{0}$.\u003c/p\u003e\n\u003ch3 id=\"example-3\"\u003eExample\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003e1. Completeness of $\\mathbb{R}^{n}$ and $\\mathbb{C}^{n}$\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u0026ndash; Consider $\\mathbb{R}^{n}$. The metric on $\\mathbb{R}^{n}$ is defined by\n$$\nd(x, y)=\\left(\\sum_{j=1}^{\\infty}\\left(x_{j}-y_{j}\\right)^{2}\\right)^{\\frac{1}{2}}\n$$\nwhere $x=\\left(x_{j}\\right)$ and $y=\\left(y_{j}\\right)$. We consider any Cauchy sequence $\\left(x_{m}\\right)$ in $\\mathbb{R}^{n}$, writing $x_{m}=\\left(x_{m, 1}, x_{m, 2}, \\ldots\\right.$, $\\left.x_{m, n}\\right)$. Since $\\left(x_{m}\\right)$ is Cauchy, for every $\\varepsilon\u0026gt;0$ there exist an $N$ such that\u003c/p\u003e\n\u003cp\u003e$$\n\\begin{equation}\n\\quad d\\left(x_{m}, x_{r}\\right)=\\left(\\sum_{j=1}^{n}\\left(x_{m, j}-x_{r, j}\\right)^{2}\\right)^{\\frac{1}{2}}\u0026lt;\\varepsilon \\quad (m, r \u0026gt; N) \\tag{1}\n\\end{equation}\n$$\u003c/p\u003e\n\u003cp\u003eSquaring, we have for $m, r\u0026gt;N$ and $j=1,2, \\ldots, n$.\n$$\n\\left(x_{m, j}-x_{r, j}\\right)^{2}\u0026lt;\\varepsilon \\quad \\text { and } \\quad\\left|x_{m, j}-x_{r, j}\\right|\u0026lt;\\varepsilon\n$$\u003c/p\u003e\n\u003cp\u003eThis show that, for each fixed $j(1 \\leqslant j \\leqslant n)$, the sequence $\\left(x_{1, j}, x_{2,j}, \\ldots\\right)$ is a Cauchy sequence of real numbers. It converges so $x_{m, j} \\rightarrow x_{j}$ as $m \\rightarrow \\infty$. Using these $n$ limits, we define $x=\\left(x_{1}, \\ldots, x_{n}\\right)$.\u003c/p\u003e\n\u003cp\u003eClearly, $x \\in \\mathbb{R}^{n}$. From (1), with $r \\rightarrow \\infty$,\n$$\nd\\left(x_{m}, x\\right)\u0026lt;\\varepsilon \\quad(m\u0026gt;N)\n$$\u003c/p\u003e\n\u003cp\u003eThis show that $x$ is the limit of $\\left(x_{m}\\right)$.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2. Completeness of $l^{\\infty}$.\u003c/strong\u003e The space $l^{\\infty}$ is complete.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3. Completeness of $c$.\u003c/strong\u003e The space $c$ consists of all convergent sequence $x=\\left(x_{j}\\right)$ of complex numbers, with the metric induced from the space $l^{\\infty}$. The space $c$ is complete.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e4. Completeness of $l^{p}$.\u003c/strong\u003e The space $l^{p}$ is complete.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e5. Completeness of $C[a, b]$.\u003c/strong\u003e The function space $C[a, b]$ is complete.\u003c/p\u003e\n\u003ch3 id=\"38-incomplete-metric-spaces\"\u003e3.8 Incomplete metric spaces\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003e1. Space $\\mathbb{Q}$\u003c/strong\u003e - set of all rational numbers\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2. Polynomials.\u003c/strong\u003e Let $X$ be the set of all polynomials considered as functions on $t$ on some finite closed interval $T=[a, b]$ and define a metric $d$ on $X$ by\n$$\nd(x, y)=\\max_{t \\in J}|x(t)-y(t)|\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3. Continuous functions.\u003c/strong\u003e Let $x$ be the set of all continuous real-valued function on $J=[0,1]$, and let\n$$\nd(x, y)=\\int_{0}^{1}|x(t)-y(t)| \\mathrm{d}t\n$$\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"4-completion-of-metric-spaces\"\u003e4. Completion of Metric Spaces\u003c/h2\u003e\n\u003ch3 id=\"41-definition-isometric-mapping-isometric-spaces\"\u003e4.1 Definition (Isometric mapping, isometric spaces).\u003c/h3\u003e\n\u003cp\u003eLet $X=(x, d)$ and $\\tilde{X}=(\\tilde{X}, \\tilde{d})$ be metric spaces. Then\u003c/p\u003e\n\u003cp\u003e$\\text{(a)} \\quad$ A mapping $T$ of $X$ into $\\tilde{X}$ is said to be isometric or an isometry if $T$ preserves distances, that is if for all $x, y \\in X$\n$$\n\\tilde{d}(T x, T y)=d(x, y)\n$$\n$\\text{(b)} \\quad$ The space $X$ is said to be isometric with the space $\\tilde{X}$ if there exists a bijective isometry of $X$ onto $\\tilde{X}.$ The space $X$ and $\\tilde{X}$ are then called isometric spaces.\u003c/p\u003e\n\u003ch3 id=\"42-theorem-completion\"\u003e4.2 Theorem (Completion)\u003c/h3\u003e\n\u003cp\u003eFor a metric space $X=(X, d)$ there exists a complete metric space $\\hat{X}=(\\hat{X}, \\hat{d})$ which has a subspace $W$ thas is isometric with $X$ and is dense in $\\hat{X}$. This space $\\hat{X}$ is unique except for isometries, that is, if $\\tilde{X}$ is any complete metric space having a dense subspace $\\tilde{W}$ isometric with $X$, then $X$ and $\\tilde{X}$ are isometric.\u003c/p\u003e\n\u003ch2 id=\"references\"\u003eReferences\u003c/h2\u003e\n\u003cp\u003e[1] Kreyszig, E. (1978). \u003cem\u003eIntroductory functional analysis with applications\u003c/em\u003e. John Wiley \u0026amp; Sons.\u003c/p\u003e\n  \u003c/div\u003e\n\u003cp\u003e\n  \u003ca href=\"mailto:nguyencungthanh11@gmail.com?subject=Re: Metric%20space\" rel='noopener noreferrer'\u003e💬 Reply via e-mail\u003c/a\u003e\n\u003c/p\u003e\n","date_modified":"2025-10-05T00:00:00+00:00","date_published":"2025-10-05T00:00:00+00:00","id":"https://nguyencungthanh.github.io/2025/10/05/metric-space/","summary":" ","tags":["Functional Analysis","Metric Space"],"title":"Metric space","url":"https://nguyencungthanh.github.io/2025/10/05/metric-space/"}],"title":"Tag: Functional Analysis - Nguyen Cung Thanh","version":"https://jsonfeed.org/version/1"}