MAST30026 Problem Sets

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

Last update: 18 March 2014

Problem Set 1

1. Check if the following functions are metrics on $X\text{.}$
   1. $d(x,y)=|x^2-y^2|$ for $x,y\in X=ℝ$
   2. $d(x,y)=|x^2-y^2|$ for $x,y\in X=(-\infty ,0]$
   3. $d(x,y)=|\text{arctan}\hspace{0.17em}x-\text{arctan}\hspace{0.17em}y|$ for $x,y\in X=ℝ$
2. (French railroad metric) Let $X={ℝ}^2$ and let $d$ be the usual metric. Denote by $\text{0}=(0,0)$ and define $$d_{\text{0}}(x,y)=\{\begin{array}{cc}0& \text{if}\hspace{0.17em}x=y\text{;}\\ d(x,\text{0})+d(\text{0},y)& \text{if}\hspace{0.17em}x\ne y\text{.}\end{array}$$ Verify that $d_0$ is a metric on $X\text{.}$ (Paris is at the origin $\text{0}\text{.)}$
3. Let $X={ℝ}^2\text{.}$ For $x=(x_1,x_2)$ and $y=(y_1,y_2)$ define $$d(x,y)=\{\begin{array}{cc}1/2& \text{if}\hspace{0.17em}{x}_1=y_1\hspace{0.17em}\text{and}\hspace{0.17em}{x}_2\ne y_2\hspace{0.17em}\text{or if}\hspace{0.17em}{x}_1\ne y_1\hspace{0.17em}\text{and}\hspace{0.17em}{x}_2=y_2\text{;}\\ 1& \text{if}\hspace{0.17em}{x}_1\ne y_1\hspace{0.17em}\text{and}\hspace{0.17em}{x}_2\ne y_2\text{;}\\ 0& \text{otherwise.}\end{array}$$ Verify that $d$ is a metric and that two congruent rectangles, one with base parallel to the $x\text{-axis}$ and the other at ${45}^{\circ }$ to the $x\text{-axis,}$ have different “area” if $d$ is used to measure the length of sides.
4. Let $(X,d)$ be a metric space. Consider the function $f:[0,\infty )\to [0,\infty )$ having the following properties:
   1. $f$ is non-decreasing, i.e. $f\left(a\right)\le f\left(b\right)$ if $0\le a<b\text{;}$
   2. $f\left(x\right)=0$ if and only if $x=0\text{;}$
   3. $f(a+b)\le f\left(a\right)+f\left(b\right),$ $a,b\in [0,\infty )\text{.}$
   If $x,y\in X$ define $d_f(x,y)=f\left(d(x,y)\right)\text{.}$ Show that $d_f$ is a metric and that the functions $f\left(t\right)=kt$ where $k>0,$ $f\left(t\right)=t^{\alpha }$ where $0<\alpha \le 1,$ and $f\left(t\right)=\frac{t}{1+t}$ for $t\ge 0$ have properties (a)–(c).
5. ($p\text{-adic}$ metric) Let $p$ be a prime number. Define the $p\text{-adic}$ absolute value function ${|·|}_p$ on $\mathbb{Q}$ by setting ${\left|x\right|}_p=0$ when $x=0$ and ${\left|x\right|}_p=p^{-k}$ when $x=p^k·\frac{m}{n}$ where $m,n$ are nonzero integers which are not divisible by $p\text{.}$ Show that for $x,y\in \mathbb{Q},$ $${|x+y|}_p\le \text{max}\{{\left|x\right|}_p,{\left|y\right|}_p\}$$ and that $d(x,y)={|x-y|}_p$ defines a metric on $\mathbb{Q}\text{.}$ In fact, $d(x,z)\le \text{max}\{d(x,y),d(y,z)\}\text{.}$ If $d$ satisfies this condition which is stronger than the triangle inequality then $d$ is called an ultrametric.
6. Let $(X_i,d_i)$ be a metric space for $1\le i\le n$ and let $X=\prod _{i=1}^n{X}_i\text{.}$ Define $$\begin{array}{ccc}d(x,y)& =& {\left[\sum _{i=1}^n{d}_i{(x_i,y_i)}^2\right]}^{1/2},\\ \bar{d}(x,y)& =& \text{max}\left\{d_i(x_i,y_i)\hspace{0.17em}\right|\hspace{0.17em}1\le i\le n\},\end{array}$$ where $x=(x_1,\dots ,x_n)$ and $y=(y_1,\dots ,y_n)\in X\text{.}$ Verify that $d$ and $\bar{d}$ are metrics on $X\text{.}$
7. Fix a positive integer $n\text{.}$ Denote by ${𝒫}_n$ the real vector space of all polynomials $p\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$ with real coefficients $a_i\text{.}$ For $p\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0\in {𝒫}_n$ set $$\Vert p\Vert =\text{max}\{\left|a_0\right|,\left|a_1\right|,\dots ,\left|a_n\right|\}$$ Verify that $\Vert ·\Vert$ is a norm on ${𝒫}_n\text{.}$
8. Let $(X_n,d_n),$ $n\in \mathbb{N},$ be a sequence of metric spaces and let $X=\prod _{n\in \mathbb{N}}{X}_n$ be the cartesian product of the $X_n\text{'s.}$ (The elements of $X$ are of the form $x=(x_1,x_2,\dots )$ with $x_n\in X_n\text{.)}$ For $x,y\in X,$ define $$d(x,y)=\sum _{n=1}^{\infty }\frac{1}{2^n}\frac{d_n(x_n,y_n)}{1+d_n(x_n,y_n)}\text{.}$$ Show that $(X,d)$ is a metric space.
9. Sketch the open ball $B(0,1)$ in the metric space $({ℝ}^3,d_i),$ where $d_i$ is defined by $$\begin{array}{ccc}{d}_1(x,y)& =& |x_1-y_1|+|x_2-y_2|+|x_3-y_3|\\ d_2(x,y)& =& \sqrt{{(x_1-y_1)}^2+{(x_2-y_2)}^2+{(x_3-y_3)}^2}\\ d_{\infty }(x,y)& =& \text{max}\{|x_1-y_1|,|x_2-y_2|,|x_3-y_3|\}\end{array}$$ for $x=(x_1,x_2,x_3)$ and $y=(y_1,y_2,y_3)\in {ℝ}^3\text{.}$
10. Set $$d(n,m)=|\frac{1}{n}-\frac{1}{m}|$$ for $n,m\in \mathbb{N}\text{.}$ Then $d$ is a metric.
    1. Let $P\subset \mathbb{N}$ be the set of positive even numbers. Find $\text{diam}\left(P\right)$ and $\text{diam}(\mathbb{N}\backslash P)$ in $(\mathbb{N},d)\text{.}$
    2. For a fixed $n\in \mathbb{N}$ find all elements of $B(2n,\frac{1}{2n})$ and $B(n,\frac{1}{2n})\text{.}$

Problem Set 2

1. Let $X={ℝ}^2\text{.}$ For $x=(x_1,x_2)$ and $y=(y_1,y_2)\in X$ define $$d_M(x,y)=\{\begin{array}{cc}|x_2-y_2|& \text{if}\hspace{0.17em}{x}_1=y_1\text{;}\\ |x_1-y_1|+\left|x_2\right|+\left|y_2\right|& \text{if}\hspace{0.17em}{x}_1\ne y_1\text{.}\end{array}$$ Also define $$d_K(x,y)=\{\begin{array}{cc}\Vert x-y\Vert & \text{if}\hspace{0.17em}x=ty\hspace{0.17em}\text{for some}\hspace{0.17em}t\in ℝ\text{;}\\ \Vert x\Vert +\Vert y\Vert & \text{otherwise}\end{array}$$ where $\Vert x\Vert ={\left[\sum _{i=1}^n{x}_i^2\right]}^{1/2}\text{.}$
   (Can you give reasonable interpretations of the metrics $d_M$ and $d_K\text{?)}$
   Study the convergence of the sequence $x_n$ in the spaces $(X,d_M)$ and $(X,d_K)$ if
   1. $x_n=(\frac{1}{n},\frac{n}{n+1})\text{;}$
   2. $x_n=(\frac{n}{n+1},\frac{n}{n+1})\text{;}$
   3. $x_n=(\frac{1}{n},\sqrt{n+1}-\sqrt{n})\text{.}$
2. Let $\left\{x_n\right\}$ and $\left\{y_n\right\}$ be sequences in a metric space $(X,d)$ such that $x_n\to n$ and $y_n\to y$ as $n\to \infty \text{.}$ Prove that $d(x_n,y_n)\to d(x,y)$ as $n\to \infty \text{.}$
3. Metrics $d$ and $\bar{d}$ defined on $X$ are called Lipschitz equivalent if there exist positive constants $m,M$ such that $$m·d(x,y)\le \bar{d}(x,y)\le M·d(x,y)$$ for all $x,y\in X\text{.}$
   1. Show that if $d$ and $\bar{d}$ are Lipschitz equivalent, then they are equivalent. Give an example of $X$ and two equivalent metrics on $X$ which are not Lipschitz equivalent.
   2. For $p\ge 1$ and $x,y\in {ℝ}^n,$ the $l^p$ metric is defined by $$d_p(x,y)={\left[\sum _{i=1}^n{|x_i-y_i|}^p\right]}^{1/p}={\Vert x-y\Vert }_p\text{.}$$ Show that if $p,q\ge 1,$ then $d_p$ and $d_q$ are Lipschitz equivalent. (Hint: compare these with $d_{\infty }(x,y)=\text{max}\{|x_1-y_1|,\dots ,|x_n-y_n|\}\text{.)}$
4. Consider the set $X=[-1,1]$ as a metric subspace of $ℝ$ with the standard metric. Let
   1. $A=\{x\in X\hspace{0.17em}|\hspace{0.17em}1/2<\left|x\right|<2\}\text{;}$
   2. $B=\{x\in X\hspace{0.17em}|\hspace{0.17em}1/2<\left|x\right|\le 2\}\text{;}$
   3. $C=\{x\in ℝ\hspace{0.17em}|\hspace{0.17em}1/2\le \left|x\right|<1\}\text{;}$
   4. $D=\{x\in ℝ\hspace{0.17em}|\hspace{0.17em}1/2\le \left|x\right|\le 1\}\text{;}$
   5. $E=\{x\in ℝ\hspace{0.17em}|\hspace{0.17em}0<\left|x\right|\le 1\hspace{0.17em}\text{and}\hspace{0.17em}1/x\notin \mathbb{Z}\}\text{.}$
   Classify the sets in (a)–(e) as open/closed in $X$ and $ℝ\text{.}$
5. Consider ${ℝ}^2$ with the standard metric. Let
   1. $A=\left\{(x,y)\hspace{0.17em}\right|\hspace{0.17em}-1<x\le 1\hspace{0.17em}\text{and}\hspace{0.17em}-1<y<1\}\text{;}$
   2. $B=\left\{(x,y)\hspace{0.17em}\right|\hspace{0.17em}xy=0\}\text{;}$
   3. $C=\left\{(x,y)\hspace{0.17em}\right|\hspace{0.17em}x\in \mathbb{Q},y\in ℝ\}\text{;}$
   4. $D=\left\{(x,y)\hspace{0.17em}\right|\hspace{0.17em}-1<x<1\hspace{0.17em}\text{and}\hspace{0.17em}y=0\}\text{;}$
   5. $E=\bigcup _{n=1}^{\infty }\left\{(x,y)\hspace{0.17em}\right|\hspace{0.17em}x=1/n\hspace{0.17em}\text{and}\hspace{0.17em}\left|y\right|\le n\}\text{.}$
   Sketch (if possible) and classify the sets in (a)–(e) as open/closed/neither in ${ℝ}^2\text{.}$
6. Find the interior, the closure and the boundary of each of the following subsets of ${ℝ}^2$ with the standard metric:
   1. $A=\left\{(x,y)\hspace{0.17em}\right|\hspace{0.17em}x>0\hspace{0.17em}\text{and}\hspace{0.17em}y\ne 0\}\text{;}$
   2. $B=\left\{(x,y)\hspace{0.17em}\right|\hspace{0.17em}x\in \mathbb{N},y\in ℝ\}\text{;}$
   3. $C=A\cup B\text{;}$
   4. $D=\left\{(x,y)\hspace{0.17em}\right|\hspace{0.17em}x\hspace{0.17em}\text{is rational}\}\text{;}$
   5. $E=\left\{(x,y)\hspace{0.17em}\right|\hspace{0.17em}x\ne 0\hspace{0.17em}\text{and}\hspace{0.17em}y\le 1/x\}\text{.}$
7. Let $A$ be a subset of a metric space $X\text{.}$ Is the interior of $A$ equal to the interior of the closure of $A\text{?}$ Is the closure of the interior of $A$ equal to the closure of $A$ itself?
8. Consider a collection ${\left\{A_i\right\}}_{i\in I}$ of subsets of a metric space $X\text{.}$ Show that $$\begin{array}{cc}\bigcup _{i\in I}{A}_i^{\circ }\subset {\left(\bigcup _{i\in I}{A}_i\right)}^{\circ }& \overline{\bigcap _{i\in I}{A}_i}\subset \bigcap _{i\in I}\overline{A_i}\\ {\left(\bigcap _{i\in I}{A}_i\right)}^{\circ }\subset \bigcap _{i\in I}{A}_i^{\circ }& \bigcup _{i\in I}\overline{A_i}\subset \overline{\bigcup _{i\in I}{A}_i}\end{array}$$
9. Let $(X,d)$ be a metric space. Show that if $A\subset X,$ then
   1. $\bar{A}=A\cup \partial A\text{.}$
   2. $\partial A=\bar{A}\backslash A^{\circ }$ and $A^{\circ }=A\backslash \partial A\text{.}$
   3. $A$ is closed if and only if $\partial A=A\backslash A^{\circ }\text{.}$
   4. $A$ is open if and only if $\partial A=\bar{A}\backslash A\text{.}$
10. Let $X$ and $Y$ be metric spaces and $A,B$ non-empty subsets of $X$ and $Y,$ respectively. Prove that
    1. If $A\times B$ is an open subset of $X\times Y,$ then $A$ and $B$ are open in $X$ and $Y,$ respectively.
    2. If $A\times B$ is an closed subset of $X\times Y,$ then $A$ and $B$ are closed in $X$ and $Y,$ respectively.

Problem Set 3

1. Let $d$ and $d\prime$ be equivalent metrics on $X\text{.}$ Show that
   1. $A\subset X$ is closed in $(X,d)$ if and only if $A$ is closed in $(X,d\prime )\text{;}$
   2. $A\subset X$ is open in $(X,d)$ if and only if $A$ is open in $(X,d\prime )\text{.}$
2. Show that if $A\subset X$ then $\text{diam}\hspace{0.17em}A=\text{diam}\hspace{0.17em}\bar{A}\text{.}$ Does $\text{diam}\hspace{0.17em}A=\text{diam}\hspace{0.17em}{A}^0\text{?}$
3. Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces and $A,B$ are dense subsets of $X$ and $Y,$ respectively. Show that $A\times B$ is dense in $X\times Y\text{.}$
4. Let $C$ be the circle in ${ℝ}^2$ with the centre at $(0,1/2)$ and radius $1/2\text{.}$ Let $X=C\backslash \left\{(0,1)\right\}\text{.}$ Define the function $f:ℝ\to X$ by defining $f\left(t\right)$ to be the point at which the line segment from $(t,0)$ to $(0,1)$ intersects $X\text{.}$
   1. Show that $f:ℝ\to X$ and $f^{-1}:X\to ℝ$ are continuous.
   2. Define for $s,t\in ℝ$ $$\rho (s,t)=\Vert f\left(s\right)-f\left(t\right)\Vert$$ where $\Vert ·\Vert$ is the standard norm in ${ℝ}^2\text{.}$ Show that $\rho$ defines a metric on $ℝ$ which is equivalent to the standard metric on $ℝ\text{.}$
5. Let $X=C[0,1]\text{.}$ Let $F:X\to ℝ$ be defined by $F\left(f\right)=f\left(0\right)\text{.}$ Moreover, let $d_{\infty }(f,g)=\text{sup}\{|f\left(x\right)-g\left(x\right)|:x\in [0,1]\}$ and $d_1(f,g)={\int }_0^1|f\left(x\right)-g\left(x\right)|dx\text{.}$
   Is $F$ continuous when $X$ is equipped with (a) the metric $d_{\infty },$ (b) the metric $d_1\text{?}$
6. Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. Show that $f:X\to Y$ is continuous if and only if
   1. $f\left(\bar{A}\right)\subset \overline{f\left(A\right)}$ for all subsets $A$ of $X,$ or
   2. $\overline{f^{-1}\left(B\right)}\subset f^{-1}\left(\bar{B}\right)$ for all subsets $B$ of $Y\text{.}$
7. Let $(X,d)$ be a metric space and let $a$ be a fixed point of $X\text{.}$ Show that $$|d(x,a)-d(y,a)|\le d(x,y)$$ for all $x,y\in X\text{.}$ Conclude that the function $f:X\to ℝ$ defined by $f\left(x\right)=d(x,a)$ is uniformly continuous.
8. Which of the following functions are uniformly continuous?
   1. $f\left(x\right)=\text{sin}\hspace{0.17em}x$ on $[0,\infty )$
   2. $g\left(x\right)=\frac{1}{1-x}$ on $(0,1)$
   3. $h\left(x\right)=\sqrt{x}$ on $[0,\infty )$
   4. $k\left(x\right)=\text{sin}(1/x),$ on $(0,1)$
9. Which of the following sequences of functions converge uniformly on the interval $[0,1]\text{?}$
   1. $f_n\left(x\right)=n{x}^2{(1-x)}^n$
   2. $f_n\left(x\right)=n^{2}x{(1-x^2)}^n$
   3. $f_n\left(x\right)=n^2{x}^3{e}^{-n{x}^2}$
10. Suppose that $A$ is a dense subset of a metric space $(X,d)$ and $f:A\to ℝ$ is uniformly continuous. Show that there exists exactly one continuous function $g:X\to ℝ$ satisfying $g\left(x\right)=f\left(x\right)$ for $x\in A\text{.}$
    (Hint: You may need to use the completeness of $ℝ\text{.)}$

Problem set 4

1. Suppose that $\left\{x_n\right\}$ and $\left\{y_n\right\}$ are Cauchy sequences in a metric space $(X,d)\text{.}$ Prove that the sequence of real numbers $\left\{d(x_n,y_n)\right\}$ converges.
2. Suppose that $\left\{x_n\right\}$ is a sequence in a metric space $(X,d)$ such that $d(x_n,x_{n+1})\le 2^{-n}$ for all $n\in \mathbb{N}\text{.}$ Prove that $\left\{x_n\right\}$ is a Cauchy sequence.
3. Decide if the following metric spaces are complete:
   1. $((0,\infty ),d),$ where $d(x,y)=|x^2-y^2|$ for $x,y\in (0,\infty )\text{.}$
   2. $((-\pi /2,\pi /2),d),$ where $d(x,y)=|\text{tan}\hspace{0.17em}x-\text{tan}\hspace{0.17em}y|$ for $x,y\in (-\pi /2,\pi /2)\text{.}$
4. Let $X=(0,1]$ be equipped with the usual metric $d(x,y)=|x-y|\text{.}$ Show that $(X,d)$ is not complete. Let $\stackrel{\sim }{d}(x,y)=|\frac{1}{x}-\frac{1}{y}|$ for $x,y\in X\text{.}$ Show that $\stackrel{\sim }{d}$ is a metric on $X$ that is equivalent to $d,$ and that $(X,\stackrel{\sim }{d})$ is complete.
5. Suppose that $(X,d)$ and $(Y,\stackrel{\sim }{d})$ are metric spaces and that $f:X\to Y$ is a bijection such that both $f$ and $f^{-1}$ are uniformly continuous. Show that $(X,d)$ is complete if and only if $(Y,\stackrel{\sim }{d})$ is complete.
6. [Cantor’s Intersection Theorem] Let $(X,d)$ be a metric space and let $\left\{F_n\right\}$ be a “decreasing” sequence of non-empty subsets of $X$ satisfying $F_{n+1}\subseteq F_n$ for all $n\text{.}$
   1. Prove that if
      1. $(X,d)$ is complete,
      2. each $F_n$ is closed,
      3. $\text{diam}\left(F_n\right)\to 0,$
      then $\bigcap _{n\in \mathbb{N}}{F}_n$ consists of exactly one point.
   2. Show that, if any of (i)-(iii) is omitted, then $\bigcap _{n\in \mathbb{N}}{F}_n$ may be empty.
   3. Conversely, prove that if for every decreasing sequence $\left\{F_n\right\}$ of non-empty subsets satisfying (ii) and (iii), the intersection $\bigcap _{n\in \mathbb{N}}{F}_n$ is non-empty, then $X$ is complete.
7. Let $(X,d)$ be a complete metric space and let $f:X\to (0,\infty )$ be a continuous function. Prove that there exists a point $x^{*}$ such that $f\left(y\right)\le 2f\left(x^{*}\right)$ for all $y\in B(x^{*},\frac{1}{\sqrt{f\left(x^{*}\right)}})\text{.}$
   (Hint: Arguing by contradiction show that there exists a sequence $\left\{x_n\right\}$ with the following properties: $f\left(x_1\right)>0,$ $f\left(x_{n+1}\right)>2f\left(x_n\right)$ for all $n\ge 1$ and $d(x_{n+1},x_n)\le \frac{1}{\sqrt{f\left(x_n\right)}}\text{.}$ Then show that $\left\{x_n\right\}$ is Cauchy.)

Problem Set 5

1. Let $\left\{f_k\right\}$ be a sequence of linear maps $f_k:{ℝ}^n\to {ℝ}^m$ which are not identically zero, that is, for every $k\in \mathbb{N}$ there is $x=x_k$ such that $f_k\left(x\right)\ne 0\text{.}$ Show that there is $x$ (not depending on $k\text{)}$ such that $f_k\left(x\right)\ne 0$ for all $k\in \mathbb{N}\text{.}$
2. Let $\left\{f_n\right\}$ be a sequence of continuous functions $f_n:ℝ\to ℝ$ having the property that $\left\{f_n\left(x\right)\right\}$ is unbounded for all $x\in \mathbb{Q}\text{.}$ Prove that there is at least one $x\in {\mathbb{Q}}^c$ such that $\left\{f_n\left(x\right)\right\}$ is unbounded.
3. Let $(X,d)$ be a complete metric space and let $(Y,\stackrel{\sim }{d})$ be a metric space. Let $\left\{f_n\right\}$ be a sequence of continuous functions from $X$ to $Y$ such that $\left\{f_n\left(x\right)\right\}$ converges for every $x\in X\text{.}$ Prove that for every $\epsilon >0$ there exist $k\in \mathbb{N}$ and a non-empty open subset $U$ of $X$ such that $\stackrel{\sim }{d}(f_n\left(x\right),f_m\left(x\right))<\epsilon$ for all $x\in U$ and all $n,m\ge k\text{.}$
4. Which of the following maps are contractions?
   1. $f:ℝ\to ℝ,$ $f\left(x\right)=e^{-x}\text{;}$
   2. $f:[0,\infty )\to [0,\infty ),$ $f\left(x\right)=e^{-x}\text{;}$
   3. $f:[0,\infty )\to [0,\infty ),$ $f\left(x\right)=e^{-e^x}\text{;}$
   4. $f:ℝ\to ℝ,$ $f\left(x\right)=\text{cos}\hspace{0.17em}x\text{;}$
   5. $f:ℝ\to ℝ,$ $f\left(x\right)=\text{cos}\left(\text{cos}\hspace{0.17em}x\right)\text{.}$
5. Consider the map $f:{ℝ}^2\to {ℝ}^2$ given by $$f(x,y)=\frac{1}{10}(8x+8y,x+y),\hspace{0.17em}(x,y)\in {ℝ}^2\text{.}$$ Recall metrics $d_1((x_1,y_1),(x_2,y_2))=|x_1-x_2|+|y_1-y_2|,$ $d_2((x_1,y_1),(x_2,y_2))={[{|x_1-x_2|}^2+{|y_1-y_2|}^2]}^{1/2}$ and $d_{\infty }((x_1,y_1),x_2,y_2)=\text{max}\{|x_1-x_2|,|y_1-y_2|\}\text{.}$ Is $f$ a contraction with respect to $d_1\text{?}$ $d_2\text{?}$ $d_{\infty }\text{?}$
6. 1. Consider $X=(0,a]$ with the usual metric and $f\left(x\right)=x^2$ for $x\in X\text{.}$ Find values of $a$ for which $f$ is a contraction and show that $f:X\to X$ does not have a fixed point.
   2. Consider $X=[1,\infty )$ with the usual metric and let $f\left(x\right)=x+\frac{1}{x}$ for $x\in X\text{.}$ Show that $f:X\to X$ and $d(f\left(x\right),f\left(y\right))<d(x,y)$ for all $x\ne y,$ but $f$ does not have a fixed point.
      Reconcile (a) and (b) with Banach fixed point theorem.
7. 7. Let $(X,d)$ be a complete metric space and $f:X\to X$ be a function such that $$d(f\left(x\right),f\left(y\right))\le \alpha d(x,y)$$ for all $x,y\in \bar{B}(x_0,r_0),$ where $0<\alpha <1$ and $d(x_0,f\left(x_0\right))\le (1-\alpha )·r_0\text{.}$ Prove that $f$ has a unique fixed point $p\in \bar{B}(x_0,r_0)\text{.}$
8. 1. Show that there is exactly one continuous function $f:[0,1]\to ℝ$ which satisfies the equation $${\left[f\left(x\right)\right]}^3-e^x{\left[f\left(x\right)\right]}^2+\frac{f\left(x\right)}{2}=e^x\text{.}$$ (Hint: rewrite the equation as $f\left(x\right)=e^x+\frac{1}{2}\frac{f\left(x\right)}{1+f{\left(x\right)}^2}\text{.)}$
   2. Consider $C[0,a]$ with $a<1$ and $T:C[0,a]\to C[0,a]$ given by $$\left(Tf\right)\left(t\right)=\text{sin}\hspace{0.17em}t+{\int }_0^{t}f\left(s\right)ds,\hspace{0.17em}t\in [0,a]\text{.}$$ Show that $T$ is a contraction. What is the fixed point of $T\text{?}$
   3. Find all $f\in C[0,\pi ]$ which satisfy the equation $$3f\left(t\right)={\int }_0^t\text{sin}(t-s)f\left(s\right)ds\text{.}$$
   4. Let $g\in C[0,1]\text{.}$ Show that there exists exactly one $f\in C[0,1]$ which solves the equation $$f\left(x\right)+{\int }_0^1{e}^{x-y-1}f\left(y\right)dy=g\left(x\right),\hspace{0.17em}\text{for all}\hspace{0.17em}x\in [0,1]\text{.}$$ (Hint: Consider the metric $d(f,h)=\text{sup}\{e^{-x}|f\left(x\right)-h\left(x\right)|:x\in [0,1]\}\text{.)}$

Problem Set 6

1. On $ℝ$ consider the metrics: $$\begin{array}{ccc}{d}_1(x,y)& =& |\text{arctan}\hspace{0.17em}x-\text{arctan}\hspace{0.17em}y|,\\ d_2(x,y)& =& |x^3-y^3|\text{.}\end{array}$$ With which of these metrics is $ℝ$ complete? If $(ℝ,d_i)$ is not complete find its completion.
2. Which of the following subsets of $ℝ$ and ${ℝ}^2$ are compact? $\text{(}ℝ$ and ${ℝ}^2$ are considered with the usual metrics).
   1. $A=\mathbb{Q}\cap [0,1]$
   2. $B=\{(x,y)\in {ℝ}^2:x^2+y^2=1\}$
   3. $C=\{(x,y)\in {ℝ}^2:x^2+y^2<1\}$
   4. $D=\{(x,y):\left|x\right|+\left|y\right|\le 1\}$
   5. $E=\{(x,y):x\ge 1\hspace{0.17em}\text{and}\hspace{0.17em}0\le y\le 1/x\}$
3. Prove that if $A_1,\dots ,A_k$ are compact subsets of a metric space $(X,d),$ then $\bigcup _{i=1}^k{A}_i$ is compact.
4. Prove that if $A_i$ is a compact subset of the metric space $(X_i,d_i)$ for $i=1,\dots ,k,$ then $A_1\times \cdots \times A_k$ is a compact subset of $X=X_1\times \cdots \times X_k$ with the product metric $d\text{.}$
5. Let $A$ be a non-empty compact subset of a metric space $(X,d)\text{.}$ Prove:
   1. If $x\in X,$ then there exists $a\in A$ such that $d(x,A)=d(x,A)\text{;}$
   2. If $A\subset U$ and $U$ is open, then there is $\epsilon >0$ such that $\{x\in X:d(x,A)<\epsilon \}\subset U\text{.}$
   3. If $B$ is closed and $A\cap B=\varnothing ,$ then $d(A,B)>0\text{.}$
   Hint: Recall that $(x,y)\mapsto d(x,y)$ is continuous from $X\times X\to [0,\infty )\text{.}$
6. Let $f:X\to ℝ\text{.}$ Call a function $f$ upper semicontinuous, abbreviated u.s.c., if for every $r\in ℝ,$ $\{x\in X\hspace{0.17em}|\hspace{0.17em}f\left(x\right)<r\}$ is open. Similarly, $f$ is lower semicontinuous, abbreviated l.s.c., if for every $r\in ℝ,$ $\{x\in X\hspace{0.17em}|\hspace{0.17em}f\left(x\right)>r\}$ is open. Assume that $X$ is compact. Show that every u.s.c. function assumes a maximum value and every l.s.c. function assumes a minimum value.
7. Call a map $f:X\to X$ a weak contraction if $d(f\left(x\right),f\left(y\right))<d(x,y)$ for all $x\ne y\text{.}$ Prove that if $X$ is compact and $f$ is a weak contraction, then $f$ has a unique fixed point.

The next problem gives a different construction of the completion of a metric space $(X,d)\text{.}$

An equivalence relation on a set $X$ is a relation $\sim$ having the following three properties:

1. (Reflexivity) $x\sim x$ for every $x\in X\text{.}$
2. (Symmetry) If $x\sim y,$ then $y\sim x\text{.}$
3. (Transitivity) If $x\sim y$ and $y\sim z,$ then $x\sim z\text{.}$

The equivalence class determined by $x,$ and denoted by $\left[x\right],$ is defined by $\left[x\right]=\{y\in X:y\sim x\}\text{.}$ We have $\left[x\right]=\left[y\right]$ if and only if $x\sim y,$ and $X$ is a disjoint union of these equivalence classes.

8. Let $(X,d)$ be a metric space and let $X^{*}$ be the set of Cauchy sequences $\text{x}=\left\{x_n\right\}$ in $(X,d)\text{.}$ Define a relation $\sim$ in $X^{*}$ by declaring $\text{x}=\left\{x_n\right\}\sim \text{y}=\left\{y_n\right\}$ to mean $d(x_n,y_n)\to 0\text{.}$
   1. Show that $\sim$ is an equivalence relation.
      Denote by $\left[\text{x}\right]$ the equivalence class of $\text{x}\in X^{*},$ and let $\stackrel{\sim }{X}$ denote the set of these equivalence classes.
   2. Show that if $\text{x}=\left\{x_n\right\}$ and $\text{y}=\left\{y_n\right\}\in X^{*},$ then $\underset{n\to \infty }{\text{lim}}d(x_n,y_n)$ exists. Show that if $\text{x}\prime =\left\{x_n^{\prime }\right\}\in \left[\text{x}\right]$ and $\text{y}\prime =\left\{y_n^{\prime }\right\}\in \left[\text{y}\right],$ then $$\underset{n\to \infty }{\text{lim}}d(x_n,y_n)=\underset{n\to \infty }{\text{lim}}d(x_n^{\prime },y_n^{\prime })\text{.}$$ For $\left[\text{x}\right],\left[\text{y}\right]\in \stackrel{\sim }{X},$ define $$D(\left[\text{x}\right],\left[\text{y}\right])=\underset{n\to \infty }{\text{lim}}d(x_n,y_n)\text{.}$$ Note that the definition of $D$ is unambiguous in view of the above equality.
   3. Show that $(\stackrel{\sim }{X},D)$ is a complete metric space.
      Hint: Let $\left[{\text{x}}^n\right]$ be Cauchy in $(\stackrel{\sim }{X},D)\text{.}$ Then ${\text{x}}^n=\{x_1^n,x_2^n,x_3^n,\dots \}$ is Cauchy in $(X,d)\text{.}$ So for every $n\in \mathbb{N},$ there exists $k_n\in \mathbb{N}$ such that $$d(x_m^n,x_{k_n}^n)<1/n$$ for all $m\ge k_n\text{.}$ Set ${\text{x}}^n=\{x_1^n,x_2^n,x_3^n,\dots \}\text{.}$ Then show that $\text{x}$ is Cauchy in $(X,d)$ and $D(\left[{\text{x}}^n\right],\left[\text{x}\right])\to 0\text{.}$
   4. If $x\in X,$ let $\phi \left(x\right)$ be the equivalence class of the constant sequence $\text{x}=(x,x,x,\dots )\text{.}$ That is, $\phi \left(x\right)=\left[\text{x}\right]=\left[\{x,x,x,\dots \}\right]\text{.}$ Show that $\phi :X\to \phi \left(X\right)$ is an isometry.
   5. Show that $\phi \left(X\right)$ is dense in $(\stackrel{\sim }{X},D)\text{.}$
      Hint: Let $\left[\text{x}\right]\in \stackrel{\sim }{X}$ with $\text{x}=\{x_1,x_2,x_3,\dots \}\text{.}$ Denote by ${\text{x}}^n$ the constant sequence $\{x_n,x_n,x_n,\dots \}$ and show that $D(\left[{\text{x}}^n\right],\left[\text{x}\right])\to 0\text{.}$

Problem Set 7

1. Consider the following spaces:
   1. $ℝ$ with the metric $d_1(x,y)=\frac{|x-y|}{1+|x-y|}\text{;}$
   2. $ℝ$ with the metric $d_2(x,y)=|\text{arctan}\hspace{0.17em}x-\text{arctan}\hspace{0.17em}y|\text{;}$
   3. $ℝ$ with the metric $d_3(x,y)=0$ if $x=y$ and $d(x,y)=1$ if $x\ne y\text{.}$
   Is $(ℝ,d_i)$ compact?
2. Use the Heine-Borel property to prove that if $f:X\to Y$ is a continuous mapping between metric spaces and $X$ is compact then $f$ is uniformly continuous.
3. A family ${\left\{F_i\right\}}_{i\in I}$ is said to have the finite intersection property if for every finite subset $J$ of $I,$ $\bigcap _{i\in J}{F}_i\ne \varnothing \text{.}$ Show that $X$ is compact if and only if for every family ${\left\{F_i\right\}}_{i\in I}$ of closed subsets of $X$ having the finite intersection property, the intersection $\bigcap _{i\in I}{F}_i\ne \varnothing \text{.}$
4. Consider $C[0,1]$ with the usual $d_{\infty }$ metric. Let $$A=\{f\in C[0,1]:0=f\left(0\right)\le f\left(t\right)\le f\left(1\right)=1\hspace{0.17em}\text{for all}\hspace{0.17em}t\in [0,1]\}\text{.}$$ Show that there is no finite $1/2\text{-net}$ for $A\text{.}$
5. Show that if $A\subset X$ is totally bounded, then $\bar{A}$ is also totally bounded.
6. Show that a metric space $(X,d)$ is totally bounded if and only if every sequence $\left\{x_n\right\}\subseteq X$ contains a Cauchy subsequence.
7. Let $X$ be a totally bounded metric space and $Y$ a metric space. Assume that $f:X\to Y$ is a bijection. Show that if $f$ and $f^{-1}$ are uniformly continuous, then $Y$ is totally bounded.
8. [Lebesgue number lemma] Let $(X,d)$ be a compact metric space and let ${\left\{U_i\right\}}_{i\in I}$ be an open covering of $X\text{.}$ Prove that there exists $\delta >0$ such that for every subset $A\subset X$ with $\text{diam}\left(A\right)<\delta$ there exists $i\in I$ such that $A\subset U_i\text{.}$
   $\text{(}\delta$ is called a “Lebesgue number” for the covering.)
9. Let $(X,d)$ be a compact metric space. Assume that $f:X\to X$ preserves distance, that is, $$d(f\left(x\right),f\left(y\right))=d(x,y)$$ for every $x,y\in X\text{.}$ Show that $f$ is a bijection.
   Hint: Assume that $f\left(X\right)\ne X\text{.}$ So there exists $a\in X\backslash f\left(X\right)\text{.}$ Since $f$ is continuous and $X$ is compact, $f\left(X\right)$ is compact. So $d(a,f\left(X\right))=r>0\text{.}$ Consider a sequence $x_n=f^n\left(a\right)\text{.}$

Problem Set 8

1. Which of the following sets $X$ are connected in ${ℝ}^2\text{?}$
   1. Let $H=\{(x,y)\in {ℝ}^2\hspace{0.17em}|\hspace{0.17em}xy=1\hspace{0.17em}\text{and}\hspace{0.17em}x,y>0\},$ $L=\left\{(x,0)\hspace{0.17em}\right|\hspace{0.17em}x\in ℝ\},$ and $X=H\cup L\text{;}$
   2. Let $C_n=\{(x,y)\in {ℝ}^2\hspace{0.17em}|\hspace{0.17em}{(x-1/n)}^2+y^2=1/n^2\}$ for $n\in \mathbb{Z},$ and $X=\bigcup _{n\in \mathbb{N}}{C}_n\text{.}$
2. Show that if $A$ is a connected subspace of a topological space $(X,𝒯)$ and if $A\subset B\subset \bar{A},$ then $B$ is connected.
3. If $A$ and $B$ are connected subsets of a topological space $(X,𝒯)$ such that $\bar{A}\cap B\ne \varnothing ,$ then $A\cup B$ is connected.
4. A point $p\in X$ is called a cut point if $X\backslash \left\{p\right\}$ is disconnected. Show that the property of having a cut point is a topological property. (A property of a topological space is a topological property if it is preserved under homeomorphisms.)
5. Show that no two of the intervals $(a,b),(a,b],$ and $[a,b]$ are homeomorphic.
6. Show that $ℝ$ and ${ℝ}^2$ are not homeomorphic (where $ℝ$ and ${ℝ}^2$ are equipped with the usual topologies).
7. Let $S^1=\{(x,y)\in {ℝ}^2\hspace{0.17em}|\hspace{0.17em}{x}^2+y^2=1\}$ be the unit circle in ${ℝ}^2,$ and let $f:S^1\to ℝ$ be a continuous function. Show that there exists $x\in S^1$ such that $f\left(x\right)=f(-x)\text{.}$ [Hint: consider the function $g:S^1\to ℝ$ where $g\left(x\right)=f\left(x\right)-f(-x)\text{.]}$
8. Let $A$ be a countable set. Show that ${ℝ}^2\backslash A$ is path connected.
9. Show that if $A$ is an open connected subset of ${ℝ}^n,$ then $A$ is path connected. [Hint: Fix a point $x_0\in A$ and consider the set $U$ of all $x\in A$ which can be joined to $x_0$ by a path in $A\text{.}$ Show that $U$ and $A\backslash U$ are open.]
10. A metric space $(X,d_X)$ is called a chain connected if for every pair $x,y$ of points in $X$ and every $\epsilon >0,$ there are finitely many points $x=x_0,x_1,x_2,\dots ,x_n=y$ such that $d_X(x_{i+1},x_i)<\epsilon$ for $i=0,1,\dots ,n-1\text{.}$ Prove that a compact, chain connected metric space is connected.

Problem Set 9

1. Show that ${ℝ}^n$ becomes a real inner product space with $⟨X,Y⟩=X^{T}AY,$ where $X,Y\in {ℝ}^n$ are column vectors and $A$ is a real symmetric matrix with positive eigenvalues. Similarly show that ${ℂ}^n$ is a complex inner product space with $⟨X,Y⟩=X^{T}BY,$ where $B$ is a Hermitian matrix with positive eigenvalues. (Recall that a matrix $B$ is Hermitian if ${\bar{B}}^T=B,$ ie $B$ is equal to the result of taking the complex conjugate transpose of $B\text{).}$
2. Show that $\Vert x\Vert =\text{sup}\frac{\left|⟨x,y⟩\right|}{\Vert y\Vert },$ over all $y\ne 0$ in any inner product space.
3. Let $H$ be the Hilbert space $L^2[-1,1]\text{.}$ Show that Gram Schmidt applied to the total set $\{1,t,t^2,t^3,\dots \}$ yields an orthonormal basis which is the sequence of Legendre polynomials given by $$L_k\left(t\right)=c_k\frac{d^k}{d{t}^k}{(t^2-1)}^k,k=1,2,3,\dots$$ where the $c_k$ are determined by requiring the polynomials to have unit length. In particular, show that the polynomials are orthogonal for any choice of $c_k\text{.}$ (You don’t need to compute the $c_k\text{).}$
4. Let $W$ be a subspace of a Hilbert space $H$ which admits an orthogonal projection $P\text{.}$ Show;
   1. $P^2=P$
   2. $\text{dist}(x,W)=\Vert x-Px\Vert ,$ ie $Px$ is the closest point to $x$ in $W\text{.}$
5. Let $A,B$ be subsets of a Hilbert space $H\text{.}$ If $A^{\perp }$ is the orthogonal complement of $A,$ defined by $A^{\perp }=\{x\in H:⟨x,a⟩=0,a\in A\}$ then prove;
   1. $A^{\perp }$ is a closed subspace of $H$
   2. $A\cap A^{\perp }\subset \left\{0\right\}$
   3. $A\subset B\Rightarrow A^{\perp }\supset B^{\perp }$
   4. $A\subset A^{\perp \perp }\text{.}$
   5. If $W$ is a subspace of $H$ then $W$ is closed if and only if $W=W^{\perp \perp }\text{.}$
6. Let $S=\{e_1,e_2,\dots \}$ be a countable orthonormal basis for a separable Hilbert space $H\text{.}$ Prove;
   1. $x=\sum _n⟨x,e_n⟩e_n$ (Fourier series)
   2. ${\Vert x\Vert }^2=\sum _n{\Vert x,e_n\Vert }^2$ (Parseval's identity for norms)
   3. $⟨x,y⟩=\sum _n⟨x,e_n⟩\overline{⟨y,e_n⟩}$ (Parseval's identiy for inner products)
7. If $W,V$ are closed subspaces of a Hilbert space $H$ and $W\perp V$ then show that $W+V$ is closed.
8. Let $S$ be a subset of a Hilbert space $H$ satisfying $S^{\perp }=\left\{0\right\}\text{.}$ Show that $S$ is a total set in $H,$ i.e $\overline{⟨S⟩}=H,$ ie the closure of the span of $S$ is $H\text{.}$
9. Let $S$ be an arbitrary set. By $l^2\left(S\right)$ we mean the set of all functions $f:S\to ℂ$ such that $f\left(s\right)\ne 0$ for countably many $s\in S$ and such that the series $\sum _{\{s\in S\}}{\left|f\left(s\right)\right|}^2$ converges. Define $⟨f,g⟩=\sum _{\{s\in S\}}⟨f\left(s\right),g\left(s\right)⟩$ for $f,g\in l^2\left(S\right)\text{.}$ Prove that;
   1. $l^2\left(S\right)$ is a Hilbert space.
   2. $l^2=l^2\left(\mathbb{N}\right)$
   3. Every Hilbert space with an orthonormal basis $S$ is isometric to $l^2\left(S\right)\text{.}$
   (Hint: define functions $f_e:S\to ℂ$ by $f_e\left(e\right)=1,$ $f_e\left(e\prime \right)=0$ for $e,e\prime \in S,$ $e\ne e\prime \text{.}$ Show that $S\prime =\{f_e:e\in S\}$ is an orthonormal basis for $l^2\left(S\right)$ and the bijection $e\to f_e$ extends to an isometry $H\to l^2\left(S\right)\text{.)}$

Problem Set 10

1. Let $R$ and $L$ be the left and right shift operators in the normed space $l^p\text{.}$ So $$R(a_1,a_2,a_3,\dots )=(0,a_1,a_2,\dots ),L(a_1,a_2,a_3,\dots )=(a_2,a_3,\dots )$$ for all $(a_1,a_2,\dots )\in l^p\text{.}$
   1. Show that $R,L$ are bounded linear operators.
   2. Show that $R$ is injective but not surjective and $L$ is surjective but not injective.
   3. Show that $LR=I$ but $RL\ne I$
   4. Show that $\Vert L^n\left(x\right)\Vert \to 0$ for all $x\in l^p$ but $\Vert L^n\Vert \nrightarrow 0\text{.}$
   5. Find the norms $\Vert L\Vert ,\Vert R\Vert \text{.}$
2. For the case $p=2$ find the adjoints of the shift operators $R,L:l^2\to l^2\text{.}$
3. Let $a$ be a fixed element of a Hilbert space $H\text{.}$ Prove that the mapping $f\left(x\right)=⟨x,a⟩$ is a bounded linear functional with $\Vert f\Vert =\Vert a\Vert \text{.}$
4. Prove the following facts about adjoints of bounded linear operators on Hilbert spaces.
   1. ${(T+S)}^{*}=T^{*}+S^{*}$
   2. ${\left(TS\right)}^{*}=S^{*}{T}^{*}$
   3. ${\left(\lambda T\right)}^{*}=\bar{\lambda }{T}^{*}$
   4. $\Vert T^{*}T\Vert ={\Vert T\Vert }^2$
5. Let $S_n$ be a sequence of self adjoint operators on a Hilbert space $H$ which converge pointwise to a bounded linear operator $S\text{.}$ Show that $S$ is self adjoint.
6. If $T$ is a positive operator on a Hilbert space $H$ prove that $T^n$ is positive for all $n\ge 1\text{.}$
7. Prove that if $T$ is a positive operator then every eigenvalue of $T$ is non-negative.
8. Let $P$ be an orthogonal projection on a Hilbert space $H\text{.}$ Prove that $P$ is self adjoint, positive and $I-P$ is positive.
9. Let $T:{ℂ}^n\to {ℂ}^m$ be a linear transformation with matrix $\left[a_{jk}\right]$ relative to the standard basis, so that the matrix is $m\times n\text{.}$ Suppose the norms in ${ℂ}^n,{ℂ}^m$ are both the supremum norm. Show that $\Vert T\Vert ={\text{max}}_j\sum _k\left|a_{jk}\right|\text{.}$
10. Let $T:{ℂ}^n\to {ℂ}^m$ be a linear transformation with matrix $\left[a_{jk}\right]$ relative to the standard basis, so that the matrix is $m\times n\text{.}$ Suppose the norms in ${ℂ}^n,{ℂ}^m$ are both the $l^1$ norms. Show that $\Vert T\Vert ={\text{max}}_k\sum _k\left|a_{jk}\right|\text{.}$

Problem Set 11

1. Let $T$ be a bounded self adjoint compact operator on a Hilbert space $H\text{.}$ If $\lambda$ is a non zero complex number so that $\lambda I-T$ is a one-to-one mapping, prove that $\lambda I-T$ is onto and has a bounded inverse. (Hint: Use the spectral theorem).
2. Let $T$ be a bounded self adjoint compact operator on a Hilbert space $H\text{.}$ If $\lambda$ is a non zero complex number so that $\lambda I-T$ is an onto mapping, prove that $\lambda I-T$ is one-to-one and has a bounded inverse. (Hint: Show that if $N$ is the null space for $\lambda I-T$ and $R$ is the closure of the range of $\lambda I-T$ then $N=R^{\perp }\text{.)}$
3. For the final example in the notes, check that the functions $s_n\left(t\right)$ are indeed eigenvectors for the Fredholm integral operator $T$ associated to the Green’s function $G$ with eigenvalues $n^2\text{.}$
4. Check that the functions $s_n\left(t\right)$ form an orthonormal set in $L^2[0,\pi ]\text{.}$
5. Check that the eigenvectors for $L$ are the solutions to the Sturm Liouville system in the last example $y″+\lambda y=0$ on $[0,\pi ]$ with the boundary conditions $y\left(0\right)=y\left(\pi \right)=0\text{.}$

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