{"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/hilbert-space/feed.json","home_page_url":"https://nguyencungthanh.github.io/tags/hilbert-space/","items":[{"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/"}],"title":"Tag: Hilbert Space - Nguyen Cung Thanh","version":"https://jsonfeed.org/version/1"}