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 13: Cauchy sequences and complete spaces

Completeness

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

A Cauchy sequence is a sequence $$\begin{array}{cccc}\stackrel{\rightharpoonup }{x}:& {\mathbb{Z}}_{>0}& ⟶& X\\ & n& ⟼& x_n\end{array}$$ such that if $\epsilon \in {ℝ}_{>0}$ then there exists $N\in {\mathbb{Z}}_{>0}$ such that
if $m,n\in {\mathbb{Z}}_{>0}$ and $m>N$ and $n>N$ then $d(x_m,x_n)<\epsilon \text{.}$

A complete metric space is a metric space $(X,d)$ such that if $\stackrel{\rightharpoonup }{x}:{\mathbb{Z}}_{>0}\to X$ is a Cauchy sequence in $X$
then there exists $x\in X$ such that $\underset{n\to \infty }{\text{lim}}{x}_n=x\text{.}$

HW: (Convergence implies Cauchy) Let $(X,d)$ be a metric space and let $\stackrel{\rightharpoonup }{x}:{\mathbb{Z}}_{>0}\to X$ be a sequence in $X\text{.}$ Show that if there exists $x\in X$ with $\underset{n\to \infty }{\text{lim}}{x}_n=x$
then $\stackrel{\rightharpoonup }{x}:{\mathbb{Z}}_{>0}\to X$ is a Cauchy sequence in $X\text{.}$

Completeness and closure

HW: Let $(X,d)$ be a metric space and let $Y\subseteq X\text{.}$ Show that if $X$ is complete and $Y$ is closed then $Y$ is complete.

HW: Let $(X,d)$ be a metric space and let $Y\subseteq X\text{.}$ Show that if $Y$ is complete then $Y$ is closed.

Examples of completeness

HW: Show that $ℝ$ is complete.

HW: Show that if $X_1,X_2,\dots ,X_m$ are complete then $X_1\times X_2\times \cdots \times X_m$ is complete.

HW: Let $X$ and $Y$ be metric spaces and let $$C_b(X,Y)=\{f:X\to Y\hspace{0.17em}|\hspace{0.17em}f\hspace{0.17em}\text{is continuous and}\hspace{0.17em}f\left(X\right)\hspace{0.17em}\text{is bounded}\}$$ with norm $\rho :C_b(X,Y)\times C_b(X,Y)\to {ℝ}_{\ge 0}$ given by $$\rho (f,g)=\text{sup}\left\{d(f\left(x\right),g\left(x\right))\hspace{0.17em}\right|\hspace{0.17em}x\in X\}\text{.}$$ Show that if $Y$ is complete then $C_b(X,Y)$ is complete.

HW: Let $(X,d)$ be a metric space. Let $$C_b\left(X\right)=\{f:X\to ℝ\hspace{0.17em}|\hspace{0.17em}f\hspace{0.17em}\text{is continuous and}\hspace{0.17em}f\left(X\right)\hspace{0.17em}\text{is bounded}\}$$ with norm $\rho :C_b\left(X\right)\times C_b\left(X\right)\to {ℝ}_{\ge 0}$ given by $$\rho (f,g)=\text{sup}\left\{d(f\left(x\right),g\left(x\right))\hspace{0.17em}\right|\hspace{0.17em}x\in X\}\text{.}$$ Show that $C_b\left(X\right)$ is complete.

Homeomorphisms and isometries

Let $(X,𝒯)$ and $(Y,ℛ)$ be topological spaces.

A homeomorphism from $X$ to $Y$ is a function $\phi :X\to Y$ such that

(a)	$\phi$ is continuous,
(b)	the inverse function ${\phi }^{-1}:Y\to X$ exists (i.e. $\phi :X\to Y$ is bijective),
(c)	${\phi }^{-1}:Y\to X$ is continuous.

Let $(X,d)$ and $(Y,\rho )$ be metric spaces.

An isometry form $X$ to $Y$ is a function $\phi :X\to Y$ such that if $x_1,x_2\in X$ then $\rho (\phi \left(x_1\right),\phi \left(x_2\right))=d(x_1,x_2)\text{.}$

HW: Show that if $\phi :X\to Y$ is an isometry then $\phi$ is injective.

HW: Show that $\phi :\mathbb{Q}\to ℝ$ is an isometry that is not surjective.

Notes and References

These are a typed copy of Lecture 13 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on August 19, 2014.

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