{"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/metric-space/feed.json","home_page_url":"https://nguyencungthanh.github.io/tags/metric-space/","items":[{"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: Metric Space - Nguyen Cung Thanh","version":"https://jsonfeed.org/version/1"}