Metric and Hilbert Spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 4 November 2014

Lecture 9: Convergences, equivalent metrics, closure

Convergence

Let $(X,d)$ be a metric space and let $x\in X\text{.}$

First definition: A function $$\begin{array}{cccc}\stackrel{\rightharpoonup }{x}:& {\mathbb{Z}}_{>0}& ⟶& X\\ & n& ⟼& x_n\end{array}$$ converges to $x$ if $\stackrel{\rightharpoonup }{x}$ satisfies: if $\epsilon \in {ℝ}_{>0}$ then there exists $N\in {\mathbb{Z}}_{>0}$ such that
if $n\in {\mathbb{Z}}_{>0}$ and $n>N$ then $d(x_n,x)<\epsilon \text{.}$ Write $\underset{n\to \infty }{\text{lim}}{x}_n=x$ if $\stackrel{\rightharpoonup }{x}$ converges to $x\text{.}$

Second definition: A function $$\begin{array}{cccc}\stackrel{\rightharpoonup }{x}:& {\mathbb{Z}}_{>0}& ⟶& X\\ & n& ⟼& x_n\end{array}$$ converges to $x$ if $\underset{n\to \infty }{\text{lim}}(x_n,x)=0\text{.}$

HW: Let $(X,d)$ be a metric space and let $x\in X\text{.}$ Let $\stackrel{\rightharpoonup }{x}:{\mathbb{Z}}_{>0}\to X$ be a function. Show that $\stackrel{\rightharpoonup }{x}$ satisfies $(*)$ if and only if $\underset{n\to \infty }{\text{lim}}d(x_n,x)=0\text{.}$

Uniqueness of limits

HW: Let $\stackrel{\rightharpoonup }{x}:{\mathbb{Z}}_{>0}\to X$ and let $x,y\in X\text{.}$ Show that if $\underset{n\to \infty }{\text{lim}}{x}_n=x$ and $\underset{n\to \infty }{\text{lim}}{x}_n=y$ then $x=y\text{.}$

Equivalent metrics

Let $X$ be a set and let $d_1:X\times X\to {ℝ}_{>0}$ and $d_2:X\times X\to {ℝ}_{>0}$ be metrics on $X\text{.}$ The metrics $d_1$ and $d_2$ are equivalent if $d_1$ and $d_2$ satisfy: If $$\begin{array}{cccc}\stackrel{\rightharpoonup }{x}:& {\mathbb{Z}}_{>0}& ⟶& X\\ & n& ⟼& x_n\end{array}$$ and $x\in X$ then $$\begin{array}{cc}\underset{n\to \infty }{\text{lim}}{d}_1(x_n,x)=0\text{if and only if}\underset{n\to \infty }{\text{lim}}{d}_2(x_n,x)=0\text{.}& (*)\end{array}$$

HW: Show that if $d_1$ and $d_2$ satisfy if $x,y\in X$ then there exist $c_1\in {ℝ}_{>0}$ and $c_2\in {ℝ}_{>0}$ such that $$d_1(x,y)\le c_1{d}_2(x,y)\text{and}{d}_2(x,y)\le c_2{d}_1(x,y)$$ then $d_1$ and $d_2$ satisfy $(*)\text{.}$

HW:

(a)	Is the "if $x,y\in X\text{"}$ in the right place of should it be after "such that".
(b)	Why isn't this statement if and only if?

Convergence and closure

Let $(X,d)$ be a metric space.

Then $X$ is a metric space with the metric space topology.

Let $A\subseteq X$ and let $\bar{A}$ be the closure of $A\text{.}$

HW: Show that $$\bar{A}=\{x\in X\hspace{0.17em}|\hspace{0.17em}\text{there exists}\hspace{0.17em}\stackrel{\rightharpoonup }{a}:{\mathbb{Z}}_{>0}\to A,n\mapsto a_n\hspace{0.17em}\text{such that}\hspace{0.17em}\underset{n\to \infty }{\text{lim}}{a}_n=x\}\text{.}$$

		Proof.
	Let $R=\{x\in X\hspace{0.17em}|\hspace{0.17em}\text{there exists}\hspace{0.17em}\stackrel{\rightharpoonup }{a}:{\mathbb{Z}}_{>0}\to A,n\mapsto a_n\hspace{0.17em}\text{such that}\hspace{0.17em}\underset{n\to \infty }{\text{lim}}{a}_n=x\}\text{.}$ To show: (a) $R\subseteq \bar{A}\text{.}$ (b) $\bar{A}\subseteq R\text{.}$ (a) To show: If $x\in R$ then $x\in \bar{A}\text{.}$ Assume $x\in R\text{.}$ To show: $x\in \bar{A}\text{.}$ We know: There exists $\stackrel{\rightharpoonup }{a}:{\mathbb{Z}}_{>0}\to A,n\mapsto a_n$ with $\underset{n\to \infty }{\text{lim}}{a}_n=x\text{.}$ To show: $x$ is a close point of $A\text{.}$ To show: If $V$ is a neighbourhood of $x$ then there exists $a\in A$ such that $a\in V\text{.}$ Assume $V$ is a neighbourhood of $x\text{.}$ Then there exists $\epsilon \in {ℝ}_{>0}$ such that $B(x,\epsilon )\subseteq V\text{.}$ To show: There exists $a\in A$ such that $a\in V\text{.}$ Let $N\in {\mathbb{Z}}_{>0}$ such that if $n\in {\mathbb{Z}}_{>0}$ and $n\ge N$ then $d(a_n,x)<\epsilon \text{.}$ Let $a=a_N\text{.}$ Then $d(a,X)=d(a_N,x)<\epsilon \text{.}$ So $a\in B(x,\epsilon )\subseteq V\text{.}$ So $x$ is a close point of $A\text{.}$ So $R\subseteq \bar{A}\text{.}$ (b) Let $x\in \bar{A}\text{.}$ To show: $x\in R\text{.}$ To show: There exists $\stackrel{\rightharpoonup }{a}:{\mathbb{Z}}_{>0}\to A,n\mapsto a_n$ with $\underset{n\to \infty }{\text{lim}}{a}_n=x\text{.}$ We know: $x$ is a close point of $A\text{.}$ Let $n\in {\mathbb{Z}}_{>0}$ and let $a_n\in A$ such that $a_n\in B(x,\frac{1}{n})\text{.}$ Let $\stackrel{\rightharpoonup }{a}:{\mathbb{Z}}_{>0}\to A$ be given by $$\stackrel{\rightharpoonup }{a}\left(n\right)=a_n\text{.}$$ To show: $\underset{n\to \infty }{\text{lim}}{a}_n=x\text{.}$ To show: If $\epsilon \in {ℝ}_{>0}$ then there exists $N\in {\mathbb{Z}}_{>0}$ such that if $n\in {\mathbb{Z}}_{>0}$ and $n>N$ then $d(a_n,x)<\epsilon \text{.}$ Assume $\epsilon \in {ℝ}_{>0}\text{.}$ Let $N\in {\mathbb{Z}}_{>0}$ be minimal such that $N>\frac{1}{\epsilon }\text{.}$ To show: If $n\in {\mathbb{Z}}_{>0}$ and $n>N$ then $d(a_n,x)<\epsilon \text{.}$ Assume $n\in {\mathbb{Z}}_{>0}$ and $n>N\text{.}$ To show: $d(a_n,x)<\epsilon \text{.}$ Since $a_n\in B(x,\frac{1}{n}),$ $$d(a_n,x)<\frac{1}{n}<\frac{1}{N}<\epsilon \text{.}$$ So $\underset{n\to \infty }{\text{lim}}{a}_n=x\text{.}$ So $x\in R\text{.}$ So $\bar{A}\subseteq R\text{.}$ $\square$

Some definitions

Let $X$ be a topological space. Let $A\subseteq X\text{.}$

The boundary of $A$ is $\partial A=\bar{A}\cap \overline{\left(A^c\right)}\text{.}$
The set $A$ is dense in $X$ if $\bar{A}=X\text{.}$
The set $A$ is nowhere dense in $X$ if ${\left(\bar{A}\right)}^{\circ }=\varnothing \text{.}$

$\mathbb{Q}$ is dense in $ℝ\text{.}$

$(0,1]$ is dense in $[0,1]\text{.}$

The boundary of $\mathbb{Q}$ in $ℝ$ is $ℝ\text{.}$

The boundary of $(0,1]$ in $ℝ$ is $\{0,1\}$ since $$\begin{array}{rcl}{\displaystyle \overline{(0,1]}\cap \overline{\left({[0,1]}^c\right)}}& =& {\displaystyle [0,1]\cap \overline{(-\infty ,0]\cup (1,\infty )}}\\ & =& {\displaystyle [0,1]\cap ((-\infty ,0]\cup [1,\infty ))}\\ & =& {\displaystyle \{0,1\}\text{.}}\end{array}$$

${\mathbb{Z}}_{>0}$ and $\mathbb{Z}$ are nowhere dense in $ℝ\text{.}$

$ℝ$ is nowhere dense in ${ℝ}^2\text{.}$

The Cantor set is nowhere dense in $[0,1]\text{.}$ The Cantor set is closed.

Notes and References

These are a typed copy of Lecture 9 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on August 12, 2014.

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