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

Assignment 1

1. Let $A$ and $B$ be bounded subsets of a metric space $(X,d)$ such that $A\cap B\ne \varnothing \text{.}$ Show that $$\text{diam}(A\cup B)\le \text{diam}\left(A\right)+\text{diam}\left(B\right)\text{.}$$ What can you say if $A$ and $B$ are disjoint?
2. Let $X=C[0,1]=\{f:[0,1]\to ℝ\hspace{0.17em}|\hspace{0.17em}f\hspace{0.17em}\text{is continuous}\}\text{.}$ The supremum metric $d_{\infty }:X\times X\to {ℝ}_{\ge 0}$ and the $L^1$ metric $d_1:X\times X\to {ℝ}_{\ge 0}$ are defined by $$\begin{array}{rcl}{\displaystyle d_{\infty }(f,g)}& =& {\displaystyle \text{sup}\left\{|f\left(x\right)-g\left(x\right)|\hspace{0.17em}\right|\hspace{0.17em}x\in [0,1]\}\text{and}}\\ {\displaystyle d_1(f,g)}& =& {\displaystyle {\int }_0^1\mid f\left(x\right)-g\left(x\right)\mid dx\text{.}}\end{array}$$ Consider the sequence $\{f_1,f_2,f_3,\dots \}$ in $X$ where $f_n\left(x\right)=n{x}^n(1-x)$ for $0\le x\le 1\text{.}$
   1. Determine whether $\left\{f_n\right\}$ converges in $(X,d_1)\text{.}$
   2. Determine whether $\left\{f_n\right\}$ converges in $(X,d_{\infty })\text{.}$
3. Let $X$ and $Y$ be topological spaces. Let $A\subseteq X$ and $B\subseteq Y\text{.}$ Show that $$\bar{A}\times \bar{B}=\overline{A\times B}\text{.}$$
4. Let $(X,d)$ be a metric space and let $A$ be a non-empty subset of $X\text{.}$ Recall that for each $x\in X,$ the distance from $x$ to $A$ is $$d(x,A)=\text{inf}\left\{d(x,a)\hspace{0.17em}\right|\hspace{0.17em}a\in A\}\text{.}$$
   1. Prove that $\bar{A}=\{x\in X\hspace{0.17em}|\hspace{0.17em}d(x,A)=0\}\text{.}$
   2. Prove that $|d(x,A)-d(y,A)|\le d(x,y)$ for all $x,y\in X\text{.}$ [Hint: first show that $d(x,A)\le d(x,y)+d(y,A)\text{.]}$
   3. Deduce the function $f:X\to ℝ$ defined by $f\left(x\right)=d(x,A)$ is continuous.
   4. Show that if $x\notin \bar{A}$ then $U=\{y\in X\hspace{0.17em}|\hspace{0.17em}d(y,A)<d(x,A)\}$ is an open set in $X$ such that $\bar{A}\subset U$ and $x\notin U\text{.}$
5. Determine whether the following sequences of functions converge uniformly.
   1. $f_n=e^{-n{x}^2},x\in [0,1]\text{;}$
   2. $g_n=e^{-x^2/n},x\in [0,1]\text{.}$
   3. $g_n=e^{-x^2/n},x\in ℝ\text{.}$
6. Let $X$ be the set of all real sequences with finitely many non-zero terms with the supremum metric: if $\text{x}=\left(x_i\right)$ and $\text{y}=\left(y_i\right)$ then $d(\text{x},\text{y})=\text{sup}\left\{|x_i-y_i|\hspace{0.17em}\right|\hspace{0.17em}i\in {\mathbb{Z}}_{>0}\}\text{.}$
   For each $n\in \mathbb{N},$ let ${\text{x}}^n=(1,1/2,1/3,\dots ,1/n,0,0,\dots )\text{.}$
   1. Show that $\left\{{\text{x}}^n\right\}$ is a Cauchy sequence in $X\text{.}$
   2. Show that $\left\{{\text{x}}^n\right\}$ does not converge to a point in $X\text{.}$ (So $X$ is not complete.)
7. Let $X$ be a nonempty set and let $(Y,d)$ be a complete metric space. Let $f:X\to Y$ be an injective function and define $$d_f(x,y)=d(f\left(x\right),f\left(y\right))$$ for $x,y\in X\text{.}$
   1. Explain briefly why $d_f$ is a metric on $X\text{.}$
   2. Show that $(X,d_f)$ is a complete metric space if $f\left(X\right)$ is a closed subset of $Y\text{.}$
8. Let $f:{ℝ}_{\ge 0}\to {ℝ}_{\ge 0}$ be given by $$f\left(x\right)=\frac{2}{2+x}\text{.}$$
   1. Show that $f$ defines a contraction mapping $f:{ℝ}_{\ge 0}\to {ℝ}_{\ge 0}\text{.}$
   2. Fix $x_0\ge 0$ and $x_{n+1}=f\left(x_n\right)$ for all $n\ge 0\text{.}$ Show that the sequence $\left\{x_n\right\}$ converges and find its limit with respect to the usual metric on $ℝ\text{.}$
9. Let $X$ be a connected topological space. Let $f:X\to ℝ$ be continuous with $f\left(X\right)\subseteq \mathbb{Q}\text{.}$ Show that $f$ is a constant function.
10. Show that $X=\{(x,y)\in {ℝ}^2\hspace{0.17em}|\hspace{0.17em}xy=0\}$ is not homeomorphic to $ℝ\text{.}$

Notes and References

These are a typed copy of Assignment 1 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces.

page history