## MAST30026 Problem Sets

Last update: 18 March 2014

## Problem Set 1

1. Check if the following functions are metrics on $X\text{.}$
1. $d\left(x,y\right)=|{x}^{2}-{y}^{2}|$ for $x,y\in X=ℝ$
2. $d\left(x,y\right)=|{x}^{2}-{y}^{2}|$ for $x,y\in X=\left(-\infty ,0\right]$
3. $d\left(x,y\right)=|\text{arctan} x-\text{arctan} y|$ for $x,y\in X=ℝ$
2. (French railroad metric) Let $X={ℝ}^{2}$ and let $d$ be the usual metric. Denote by $\text{0}=\left(0,0\right)$ and define $d0(x,y)= { 0 if x=y; d(x,0)+ d(0,y) if x≠y.$ 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 $x=\left({x}_{1},{x}_{2}\right)$ and $y=\left({y}_{1},{y}_{2}\right)$ define $d(x,y)= { 1/2 if x1=y1 and x2≠y2 or if x1≠y1 and x2=y2; 1 if x1≠y1 and x2≠y2; 0 otherwise.$ 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 $\left(X,d\right)$ be a metric space. Consider the function $f:\left[0,\infty \right)\to \left[0,\infty \right)$ having the following properties:
1. $f$ is non-decreasing, i.e. $f\left(a\right)\le f\left(b\right)$ if $0\le a
2. $f\left(x\right)=0$ if and only if $x=0\text{;}$
3. $f\left(a+b\right)\le f\left(a\right)+f\left(b\right),$ $a,b\in \left[0,\infty \right)\text{.}$
If $x,y\in X$ define ${d}_{f}\left(x,y\right)=f\left(d\left(x,y\right)\right)\text{.}$ 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 $ℚ$ by setting ${|x|}_{p}=0$ when $x=0$ and ${|x|}_{p}={p}^{-k}$ when $x={p}^{k}·\frac{m}{n}$ where $m,n$ are nonzero integers which are not divisible by $p\text{.}$ Show that for $x,y\in ℚ,$ $|x+y|p≤ max{|x|p,|y|p}$ and that $d\left(x,y\right)={|x-y|}_{p}$ defines a metric on $ℚ\text{.}$ In fact, $d\left(x,z\right)\le \text{max}\left\{d\left(x,y\right),d\left(y,z\right)\right\}\text{.}$ If $d$ satisfies this condition which is stronger than the triangle inequality then $d$ is called an ultrametric.
6. Let $\left({X}_{i},{d}_{i}\right)$ be a metric space for $1\le i\le n$ and let $X=\prod _{i=1}^{n}{X}_{i}\text{.}$ Define $d(x,y) = [∑i=1ndi(xi,yi)2]1/2, d‾(x,y) = max { di(xi,yi) | 1≤i≤n } ,$ where $x=\left({x}_{1},\dots ,{x}_{n}\right)$ and $y=\left({y}_{1},\dots ,{y}_{n}\right)\in X\text{.}$ Verify that $d$ and $\stackrel{‾}{d}$ 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 $‖p‖=max { |a0|, |a1|,…, |an| }$ Verify that $‖·‖$ is a norm on ${𝒫}_{n}\text{.}$
8. Let $\left({X}_{n},{d}_{n}\right),$ $n\in ℕ,$ be a sequence of metric spaces and let $X=\prod _{n\in ℕ}{X}_{n}$ be the cartesian product of the ${X}_{n}\text{'s.}$ (The elements of $X$ are of the form $x=\left({x}_{1},{x}_{2},\dots \right)$ with ${x}_{n}\in {X}_{n}\text{.)}$ For $x,y\in X,$ define $d(x,y)= ∑n=1∞ 12n dn(xn,yn) 1+dn(xn,yn) .$ Show that $\left(X,d\right)$ is a metric space.
9. Sketch the open ball $B\left(0,1\right)$ in the metric space $\left({ℝ}^{3},{d}_{i}\right),$ where ${d}_{i}$ is defined by $d1(x,y) = |x1-y1|+ |x2-y2|+ |x3-y3| d2(x,y) = (x1-y1)2+ (x2-y2)2+ (x3-y3)2 d∞(x,y) = max { |x1-y1|, |x2-y2|, |x3-y3| }$ for $x=\left({x}_{1},{x}_{2},{x}_{3}\right)$ and $y=\left({y}_{1},{y}_{2},{y}_{3}\right)\in {ℝ}^{3}\text{.}$
10. Set $d(n,m)= |1n-1m|$ for $n,m\in ℕ\text{.}$ Then $d$ is a metric.
1. Let $P\subset ℕ$ be the set of positive even numbers. Find $\text{diam}\left(P\right)$ and $\text{diam}\left(ℕ\P\right)$ in $\left(ℕ,d\right)\text{.}$
2. For a fixed $n\in ℕ$ find all elements of $B\left(2n,\frac{1}{2n}\right)$ and $B\left(n,\frac{1}{2n}\right)\text{.}$

## Problem Set 2

1. Let $X={ℝ}^{2}\text{.}$ For $x=\left({x}_{1},{x}_{2}\right)$ and $y=\left({y}_{1},{y}_{2}\right)\in X$ define $dM(x,y)= { |x2-y2| if x1=y1; |x1-y1|+ |x2|+ |y2| if x1≠y1.$ Also define $dK(x,y)= { ‖x-y‖ if x=ty for some t∈ℝ; ‖x‖+‖y‖ otherwise$ where $‖x‖={\left[\sum _{i=1}^{n}{x}_{i}^{2}\right]}^{1/2}\text{.}$
(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 $\left(X,{d}_{M}\right)$ and $\left(X,{d}_{K}\right)$ if
1. ${x}_{n}=\left(\frac{1}{n},\frac{n}{n+1}\right)\text{;}$
2. ${x}_{n}=\left(\frac{n}{n+1},\frac{n}{n+1}\right)\text{;}$
3. ${x}_{n}=\left(\frac{1}{n},\sqrt{n+1}-\sqrt{n}\right)\text{.}$
2. Let $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ be sequences in a metric space $\left(X,d\right)$ such that ${x}_{n}\to n$ and ${y}_{n}\to y$ as $n\to \infty \text{.}$ Prove that $d\left({x}_{n},{y}_{n}\right)\to d\left(x,y\right)$ as $n\to \infty \text{.}$
3. Metrics $d$ and $\stackrel{‾}{d}$ defined on $X$ are called Lipschitz equivalent if there exist positive constants $m,M$ such that $m·d(x,y)≤ d‾(x,y) ≤M·d(x,y)$ for all $x,y\in X\text{.}$
1. Show that if $d$ and $\stackrel{‾}{d}$ 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 $dp(x,y)= [∑i=1n|xi-yi|p]1/p =‖x-y‖p.$ Show that if $p,q\ge 1,$ then ${d}_{p}$ and ${d}_{q}$ are Lipschitz equivalent. (Hint: compare these with ${d}_{\infty }\left(x,y\right)=\text{max}\left\{|{x}_{1}-{y}_{1}|,\dots ,|{x}_{n}-{y}_{n}|\right\}\text{.)}$
4. Consider the set $X=\left[-1,1\right]$ as a metric subspace of $ℝ$ with the standard metric. Let
1. $A=\left\{x\in X | 1/2<|x|<2\right\}\text{;}$
2. $B=\left\{x\in X | 1/2<|x|\le 2\right\}\text{;}$
3. $C=\left\{x\in ℝ | 1/2\le |x|<1\right\}\text{;}$
4. $D=\left\{x\in ℝ | 1/2\le |x|\le 1\right\}\text{;}$
5. $E=\left\{x\in ℝ | 0<|x|\le 1 \text{and} 1/x\notin ℤ\right\}\text{.}$
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\{\left(x,y\right) | -1
2. $B=\left\{\left(x,y\right) | xy=0\right\}\text{;}$
3. $C=\left\{\left(x,y\right) | x\in ℚ,y\in ℝ\right\}\text{;}$
4. $D=\left\{\left(x,y\right) | -1
5. $E=\bigcup _{n=1}^{\infty }\left\{\left(x,y\right) | x=1/n \text{and} |y|\le n\right\}\text{.}$
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. $A=\left\{\left(x,y\right) | x>0 \text{and} y\ne 0\right\}\text{;}$
2. $B=\left\{\left(x,y\right) | x\in ℕ,y\in ℝ\right\}\text{;}$
3. $C=A\cup B\text{;}$
4. $D=\left\{\left(x,y\right) | x \text{is rational}\right\}\text{;}$
5. $E=\left\{\left(x,y\right) | x\ne 0 \text{and} y\le 1/x\right\}\text{.}$
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 $⋃i∈IAi∘⊂ (⋃i∈IAi)∘ ⋂i∈IAi‾⊂ ⋂i∈IAi‾ (⋂i∈IAi)∘⊂ ⋂i∈IAi∘ ⋃i∈IAi‾⊂ ⋃i∈IAi‾$
9. Let $\left(X,d\right)$ be a metric space. Show that if $A\subset X,$ then
1. $\stackrel{‾}{A}=A\cup \partial A\text{.}$
2. $\partial A=\stackrel{‾}{A}\{A}^{\circ }$ and ${A}^{\circ }=A\\partial A\text{.}$
3. $A$ is closed if and only if $\partial A=A\{A}^{\circ }\text{.}$
4. $A$ is open if and only if $\partial A=\stackrel{‾}{A}\A\text{.}$
10. Let $X$ and $Y$ be metric spaces and $A,B$ non-empty subsets of $X$ and $Y,$ respectively. Prove that
1. If $A×B$ is an open subset of $X×Y,$ then $A$ and $B$ are open in $X$ and $Y,$ respectively.
2. If $A×B$ is an closed subset of $X×Y,$ 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 $\left(X,d\right)$ if and only if $A$ is closed in $\left(X,d\prime \right)\text{;}$
2. $A\subset X$ is open in $\left(X,d\right)$ if and only if $A$ is open in $\left(X,d\prime \right)\text{.}$
2. Show that if $A\subset X$ then $\text{diam} A=\text{diam} \stackrel{‾}{A}\text{.}$ Does $\text{diam} A=\text{diam} {A}^{0}\text{?}$
3. Let $\left(X,{d}_{X}\right)$ and $\left(Y,{d}_{Y}\right)$ be metric spaces and $A,B$ are dense subsets of $X$ and $Y,$ respectively. Show that $A×B$ is dense in $X×Y\text{.}$
4. Let $C$ be the circle in ${ℝ}^{2}$ with the centre at $\left(0,1/2\right)$ and radius $1/2\text{.}$ Let $X=C\\left\{\left(0,1\right)\right\}\text{.}$ Define the function $f:ℝ\to X$ by defining $f\left(t\right)$ to be the point at which the line segment from $\left(t,0\right)$ to $\left(0,1\right)$ intersects $X\text{.}$
1. Show that $f:ℝ\to X$ and ${f}^{-1}:X\to ℝ$ are continuous.
2. Define for $s,t\in ℝ$ $ρ(s,t)= ‖f(s)-f(t)‖$ where $‖·‖$ 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 $X=C\left[0,1\right]\text{.}$ Let $F:X\to ℝ$ be defined by $F\left(f\right)=f\left(0\right)\text{.}$ Moreover, let ${d}_{\infty }\left(f,g\right)=\text{sup}\left\{|f\left(x\right)-g\left(x\right)|:x\in \left[0,1\right]\right\}$ and ${d}_{1}\left(f,g\right)={\int }_{0}^{1}|f\left(x\right)-g\left(x\right)|dx\text{.}$
Is $F$ continuous when $X$ is equipped with (a) the metric ${d}_{\infty },$ (b) the metric ${d}_{1}\text{?}$
6. Let $\left(X,{d}_{X}\right)$ and $\left(Y,{d}_{Y}\right)$ be metric spaces. Show that $f:X\to Y$ is continuous if and only if
1. $f\left(\stackrel{‾}{A}\right)\subset \stackrel{‾}{f\left(A\right)}$ for all subsets $A$ of $X,$ or
2. $\stackrel{‾}{{f}^{-1}\left(B\right)}\subset {f}^{-1}\left(\stackrel{‾}{B}\right)$ for all subsets $B$ of $Y\text{.}$
7. Let $\left(X,d\right)$ be a metric space and let $a$ be a fixed point of $X\text{.}$ Show that $|d(x,a)-d(y,a)| ≤d(x,y)$ for all $x,y\in X\text{.}$ Conclude that the function $f:X\to ℝ$ defined by $f\left(x\right)=d\left(x,a\right)$ is uniformly continuous.
8. Which of the following functions are uniformly continuous?
1. $f\left(x\right)=\text{sin} x$ on $\left[0,\infty \right)$
2. $g\left(x\right)=\frac{1}{1-x}$ on $\left(0,1\right)$
3. $h\left(x\right)=\sqrt{x}$ on $\left[0,\infty \right)$
4. $k\left(x\right)=\text{sin}\left(1/x\right),$ on $\left(0,1\right)$
9. Which of the following sequences of functions converge uniformly on the interval $\left[0,1\right]\text{?}$
1. ${f}_{n}\left(x\right)=n{x}^{2}{\left(1-x\right)}^{n}$
2. ${f}_{n}\left(x\right)={n}^{2}x{\left(1-{x}^{2}\right)}^{n}$
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 $\left(X,d\right)$ 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 $\left(X,d\right)\text{.}$ Prove that the sequence of real numbers $\left\{d\left({x}_{n},{y}_{n}\right)\right\}$ converges.
2. Suppose that $\left\{{x}_{n}\right\}$ is a sequence in a metric space $\left(X,d\right)$ such that $d\left({x}_{n},{x}_{n+1}\right)\le {2}^{-n}$ for all $n\in ℕ\text{.}$ Prove that $\left\{{x}_{n}\right\}$ is a Cauchy sequence.
3. Decide if the following metric spaces are complete:
1. $\left(\left(0,\infty \right),d\right),$ where $d\left(x,y\right)=|{x}^{2}-{y}^{2}|$ for $x,y\in \left(0,\infty \right)\text{.}$
2. $\left(\left(-\pi /2,\pi /2\right),d\right),$ where $d\left(x,y\right)=|\text{tan} x-\text{tan} y|$ for $x,y\in \left(-\pi /2,\pi /2\right)\text{.}$
4. Let $X=\left(0,1\right]$ be equipped with the usual metric $d\left(x,y\right)=|x-y|\text{.}$ Show that $\left(X,d\right)$ is not complete. Let $\stackrel{\sim }{d}\left(x,y\right)=|\frac{1}{x}-\frac{1}{y}|$ for $x,y\in X\text{.}$ Show that $\stackrel{\sim }{d}$ is a metric on $X$ that is equivalent to $d,$ and that $\left(X,\stackrel{\sim }{d}\right)$ is complete.
5. Suppose that $\left(X,d\right)$ and $\left(Y,\stackrel{\sim }{d}\right)$ are metric spaces and that $f:X\to Y$ is a bijection such that both $f$ and ${f}^{-1}$ are uniformly continuous. Show that $\left(X,d\right)$ is complete if and only if $\left(Y,\stackrel{\sim }{d}\right)$ is complete.
6. [Cantor’s Intersection Theorem] Let $\left(X,d\right)$ 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. $\left(X,d\right)$ is complete,
2. each ${F}_{n}$ is closed,
3. $\text{diam}\left({F}_{n}\right)\to 0,$
then $\bigcap _{n\in ℕ}{F}_{n}$ consists of exactly one point.
2. Show that, if any of (i)-(iii) is omitted, then $\bigcap _{n\in ℕ}{F}_{n}$ 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 $\bigcap _{n\in ℕ}{F}_{n}$ is non-empty, then $X$ is complete.
7. Let $\left(X,d\right)$ be a complete metric space and let $f:X\to \left(0,\infty \right)$ be a continuous function. Prove that there exists a point ${x}^{*}$ such that $f\left(y\right)\le 2f\left({x}^{*}\right)$ for all $y\in B\left({x}^{*},\frac{1}{\sqrt{f\left({x}^{*}\right)}}\right)\text{.}$
(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 $d\left({x}_{n+1},{x}_{n}\right)\le \frac{1}{\sqrt{f\left({x}_{n}\right)}}\text{.}$ 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 $k\in ℕ$ 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 $k\in ℕ\text{.}$
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 $x\in ℚ\text{.}$ Prove that there is at least one $x\in {ℚ}^{c}$ such that $\left\{{f}_{n}\left(x\right)\right\}$ is unbounded.
3. Let $\left(X,d\right)$ be a complete metric space and let $\left(Y,\stackrel{\sim }{d}\right)$ 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 $k\in ℕ$ and a non-empty open subset $U$ of $X$ such that $\stackrel{\sim }{d}\left({f}_{n}\left(x\right),{f}_{m}\left(x\right)\right)<\epsilon$ 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. $f:\left[0,\infty \right)\to \left[0,\infty \right),$ $f\left(x\right)={e}^{-x}\text{;}$
3. $f:\left[0,\infty \right)\to \left[0,\infty \right),$ $f\left(x\right)={e}^{-{e}^{x}}\text{;}$
4. $f:ℝ\to ℝ,$ $f\left(x\right)=\text{cos} x\text{;}$
5. $f:ℝ\to ℝ,$ $f\left(x\right)=\text{cos}\left(\text{cos} x\right)\text{.}$
5. Consider the map $f:{ℝ}^{2}\to {ℝ}^{2}$ given by $f(x,y)=110 (8x+8y,x+y), (x,y)∈ℝ2.$ Recall metrics ${d}_{1}\left(\left({x}_{1},{y}_{1}\right),\left({x}_{2},{y}_{2}\right)\right)=|{x}_{1}-{x}_{2}|+|{y}_{1}-{y}_{2}|,$ ${d}_{2}\left(\left({x}_{1},{y}_{1}\right),\left({x}_{2},{y}_{2}\right)\right)={\left[{|{x}_{1}-{x}_{2}|}^{2}+{|{y}_{1}-{y}_{2}|}^{2}\right]}^{1/2}$ and ${d}_{\infty }\left(\left({x}_{1},{y}_{1}\right),{x}_{2},{y}_{2}\right)=\text{max}\left\{|{x}_{1}-{x}_{2}|,|{y}_{1}-{y}_{2}|\right\}\text{.}$ Is $f$ a contraction with respect to ${d}_{1}\text{?}$ ${d}_{2}\text{?}$ ${d}_{\infty }\text{?}$
1. Consider $X=\left(0,a\right]$ 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 $X=\left[1,\infty \right)$ 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\left(f\left(x\right),f\left(y\right)\right) for all $x\ne y,$ but $f$ does not have a fixed point.
Reconcile (a) and (b) with Banach fixed point theorem.
6. 7. Let $\left(X,d\right)$ be a complete metric space and $f:X\to X$ be a function such that $d(f(x),f(y)) ≤αd(x,y)$ for all $x,y\in \stackrel{‾}{B}\left({x}_{0},{r}_{0}\right),$ where $0<\alpha <1$ and $d\left({x}_{0},f\left({x}_{0}\right)\right)\le \left(1-\alpha \right)·{r}_{0}\text{.}$ Prove that $f$ has a unique fixed point $p\in \stackrel{‾}{B}\left({x}_{0},{r}_{0}\right)\text{.}$
1. Show that there is exactly one continuous function $f:\left[0,1\right]\to ℝ$ which satisfies the equation $[f(x)]3-ex [f(x)]2+ f(x)2=ex.$ (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 $C\left[0,a\right]$ with $a<1$ and $T:C\left[0,a\right]\to C\left[0,a\right]$ given by $(Tf)(t)=sin t+ ∫0tf(s)ds, t ∈[0,a].$ Show that $T$ is a contraction. What is the fixed point of $T\text{?}$
3. Find all $f\in C\left[0,\pi \right]$ which satisfy the equation $3f(t)=∫0tsin (t-s)f(s)ds.$
4. Let $g\in C\left[0,1\right]\text{.}$ Show that there exists exactly one $f\in C\left[0,1\right]$ which solves the equation $f(x)+∫01 ex-y-1f(y) dy=g(x), for all x∈[0,1].$ (Hint: Consider the metric $d\left(f,h\right)=\text{sup}\left\{{e}^{-x}|f\left(x\right)-h\left(x\right)|:x\in \left[0,1\right]\right\}\text{.)}$

## Problem Set 6

1. On $ℝ$ consider the metrics: $d1(x,y) = |arctan x-arctan y|, d2(x,y) = |x3-y3|.$ With which of these metrics is $ℝ$ complete? If $\left(ℝ,{d}_{i}\right)$ 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. $A=ℚ\cap \left[0,1\right]$
2. $B=\left\{\left(x,y\right)\in {ℝ}^{2}:{x}^{2}+{y}^{2}=1\right\}$
3. $C=\left\{\left(x,y\right)\in {ℝ}^{2}:{x}^{2}+{y}^{2}<1\right\}$
4. $D=\left\{\left(x,y\right):|x|+|y|\le 1\right\}$
5. $E=\left\{\left(x,y\right):x\ge 1 \text{and} 0\le y\le 1/x\right\}$
3. Prove that if ${A}_{1},\dots ,{A}_{k}$ are compact subsets of a metric space $\left(X,d\right),$ then $\bigcup _{i=1}^{k}{A}_{i}$ is compact.
4. Prove that if ${A}_{i}$ is a compact subset of the metric space $\left({X}_{i},{d}_{i}\right)$ for $i=1,\dots ,k,$ then ${A}_{1}×\cdots ×{A}_{k}$ is a compact subset of $X={X}_{1}×\cdots ×{X}_{k}$ with the product metric $d\text{.}$
5. Let $A$ be a non-empty compact subset of a metric space $\left(X,d\right)\text{.}$ Prove:
1. If $x\in X,$ then there exists $a\in A$ such that $d\left(x,A\right)=d\left(x,A\right)\text{;}$
2. If $A\subset U$ and $U$ is open, then there is $\epsilon >0$ such that $\left\{x\in X:d\left(x,A\right)<\epsilon \right\}\subset U\text{.}$
3. If $B$ is closed and $A\cap B=\varnothing ,$ then $d\left(A,B\right)>0\text{.}$
Hint: Recall that $\left(x,y\right)↦d\left(x,y\right)$ is continuous from $X×X\to \left[0,\infty \right)\text{.}$
6. Let $f:X\to ℝ\text{.}$ Call a function $f$ upper semicontinuous, abbreviated u.s.c., if for every $r\in ℝ,$ $\left\{x\in X | f\left(x\right) is open. Similarly, $f$ is lower semicontinuous, abbreviated l.s.c., if for every $r\in ℝ,$ $\left\{x\in X | f\left(x\right)>r\right\}$ 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\left(f\left(x\right),f\left(y\right)\right) 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 $\left(X,d\right)\text{.}$

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 $\left[x\right]=\left\{y\in X:y\sim x\right\}\text{.}$ 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.

1. Let $\left(X,d\right)$ be a metric space and let ${X}^{*}$ be the set of Cauchy sequences $\text{x}=\left\{{x}_{n}\right\}$ in $\left(X,d\right)\text{.}$ Define a relation $\sim$ in ${X}^{*}$ by declaring $\text{x}=\left\{{x}_{n}\right\}\sim \text{y}=\left\{{y}_{n}\right\}$ to mean $d\left({x}_{n},{y}_{n}\right)\to 0\text{.}$
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 $\underset{n\to \infty }{\text{lim}}d\left({x}_{n},{y}_{n}\right)$ 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 $limn→∞d (xn,yn)= limn→∞d (xn′,yn′).$ For $\left[\text{x}\right],\left[\text{y}\right]\in \stackrel{\sim }{X},$ define $D([x],[y]) =limn→∞d (xn,yn).$ Note that the definition of $D$ is unambiguous in view of the above equality.
3. Show that $\left(\stackrel{\sim }{X},D\right)$ is a complete metric space.
Hint: Let $\left[{\text{x}}^{n}\right]$ be Cauchy in $\left(\stackrel{\sim }{X},D\right)\text{.}$ Then ${\text{x}}^{n}=\left\{{x}_{1}^{n},{x}_{2}^{n},{x}_{3}^{n},\dots \right\}$ is Cauchy in $\left(X,d\right)\text{.}$ So for every $n\in ℕ,$ there exists ${k}_{n}\in ℕ$ such that $d(xmn,xknn) <1/n$ for all $m\ge {k}_{n}\text{.}$ Set ${\text{x}}^{n}=\left\{{x}_{1}^{n},{x}_{2}^{n},{x}_{3}^{n},\dots \right\}\text{.}$ Then show that $\text{x}$ is Cauchy in $\left(X,d\right)$ and $D\left(\left[{\text{x}}^{n}\right],\left[\text{x}\right]\right)\to 0\text{.}$
4. If $x\in X,$ let $\phi \left(x\right)$ be the equivalence class of the constant sequence $\text{x}=\left(x,x,x,\dots \right)\text{.}$ That is, $\phi \left(x\right)=\left[\text{x}\right]=\left[\left\{x,x,x,\dots \right\}\right]\text{.}$ Show that $\phi :X\to \phi \left(X\right)$ is an isometry.
5. Show that $\phi \left(X\right)$ is dense in $\left(\stackrel{\sim }{X},D\right)\text{.}$
Hint: Let $\left[\text{x}\right]\in \stackrel{\sim }{X}$ with $\text{x}=\left\{{x}_{1},{x}_{2},{x}_{3},\dots \right\}\text{.}$ Denote by ${\text{x}}^{n}$ the constant sequence $\left\{{x}_{n},{x}_{n},{x}_{n},\dots \right\}$ and show that $D\left(\left[{\text{x}}^{n}\right],\left[\text{x}\right]\right)\to 0\text{.}$

## Problem Set 7

1. Consider the following spaces:
1. $ℝ$ with the metric ${d}_{1}\left(x,y\right)=\frac{|x-y|}{1+|x-y|}\text{;}$
2. $ℝ$ with the metric ${d}_{2}\left(x,y\right)=|\text{arctan} x-\text{arctan} y|\text{;}$
3. $ℝ$ with the metric ${d}_{3}\left(x,y\right)=0$ if $x=y$ and $d\left(x,y\right)=1$ if $x\ne y\text{.}$
Is $\left(ℝ,{d}_{i}\right)$ 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 $C\left[0,1\right]$ with the usual ${d}_{\infty }$ metric. Let $A= { f∈C[0,1]:0=f (0)≤f(t)≤f (1)=1 for all t∈ [0,1] } .$ Show that there is no finite $1/2\text{-net}$ for $A\text{.}$
5. Show that if $A\subset X$ is totally bounded, then $\stackrel{‾}{A}$ is also totally bounded.
6. Show that a metric space $\left(X,d\right)$ 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 $\left(X,d\right)$ 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 $\left(X,d\right)$ be a compact metric space. Assume that $f:X\to X$ preserves distance, that is, $d(f(x),f(y)) =d(x,y)$ 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 $a\in X\f\left(X\right)\text{.}$ Since $f$ is continuous and $X$ is compact, $f\left(X\right)$ is compact. So $d\left(a,f\left(X\right)\right)=r>0\text{.}$ 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 $H=\left\{\left(x,y\right)\in {ℝ}^{2} | xy=1 \text{and} x,y>0\right\},$ $L=\left\{\left(x,0\right) | x\in ℝ\right\},$ and $X=H\cup L\text{;}$
2. Let ${C}_{n}=\left\{\left(x,y\right)\in {ℝ}^{2} | {\left(x-1/n\right)}^{2}+{y}^{2}=1/{n}^{2}\right\}$ for $n\in ℤ,$ and $X=\bigcup _{n\in ℕ}{C}_{n}\text{.}$
2. Show that if $A$ is a connected subspace of a topological space $\left(X,𝒯\right)$ and if $A\subset B\subset \stackrel{‾}{A},$ then $B$ is connected.
3. If $A$ and $B$ are connected subsets of a topological space $\left(X,𝒯\right)$ such that $\stackrel{‾}{A}\cap B\ne \varnothing ,$ then $A\cup B$ is connected.
4. A point $p\in X$ is called a cut point if $X\\left\{p\right\}$ 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 $\left(a,b\right),\left(a,b\right],$ and $\left[a,b\right]$ 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 ${S}^{1}=\left\{\left(x,y\right)\in {ℝ}^{2} | {x}^{2}+{y}^{2}=1\right\}$ 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 $f\left(x\right)=f\left(-x\right)\text{.}$ [Hint: consider the function $g:{S}^{1}\to ℝ$ where $g\left(x\right)=f\left(x\right)-f\left(-x\right)\text{.]}$
8. Let $A$ be a countable set. Show that ${ℝ}^{2}\A$ 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 $A\U$ are open.]
10. A metric space $\left(X,{d}_{X}\right)$ 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 ${d}_{X}\left({x}_{i+1},{x}_{i}\right)<\epsilon$ 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 ${\stackrel{‾}{B}}^{T}=B,$ ie $B$ is equal to the result of taking the complex conjugate transpose of $B\text{).}$
2. Show that $‖x‖=\text{sup}\frac{|⟨x,y⟩|}{‖y‖},$ over all $y\ne 0$ in any inner product space.
3. Let $H$ be the Hilbert space ${L}^{2}\left[-1,1\right]\text{.}$ Show that Gram Schmidt applied to the total set $\left\{1,t,{t}^{2},{t}^{3},\dots \right\}$ yields an orthonormal basis which is the sequence of Legendre polynomials given by $Lk(t)=ck dkdtk (t2-1)k, k=1,2,3,…$ 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. $\text{dist}\left(x,W\right)=‖x-Px‖,$ 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 ${A}^{\perp }=\left\{x\in H:⟨x,a⟩=0,a\in A\right\}$ then prove;
1. ${A}^{\perp }$ is a closed subspace of $H$
2. $A\cap {A}^{\perp }\subset \left\{0\right\}$
3. $A\subset B⇒{A}^{\perp }\supset {B}^{\perp }$
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 $S=\left\{{e}_{1},{e}_{2},\dots \right\}$ 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. ${‖x‖}^{2}=\sum _{n}{‖x,{e}_{n}‖}^{2}$ (Parseval's identity for norms)
3. $⟨x,y⟩=\sum _{n}⟨x,{e}_{n}⟩\stackrel{‾}{⟨y,{e}_{n}⟩}$ (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 $\stackrel{‾}{⟨S⟩}=H,$ 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 $\sum _{\left\{s\in S\right\}}{|f\left(s\right)|}^{2}$ converges. Define $⟨f,g⟩=\sum _{\left\{s\in S\right\}}⟨f\left(s\right),g\left(s\right)⟩$ for $f,g\in {l}^{2}\left(S\right)\text{.}$ Prove that;
1. ${l}^{2}\left(S\right)$ is a Hilbert space.
2. ${l}^{2}={l}^{2}\left(ℕ\right)$
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 $S\prime =\left\{{f}_{e}:e\in S\right\}$ 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 $R(a1,a2,a3,…)= (0,a1,a2,…), L(a1,a2,a3,…)= (a2,a3,…)$ for all $\left({a}_{1},{a}_{2},\dots \right)\in {l}^{p}\text{.}$
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 $‖{L}^{n}\left(x\right)‖\to 0$ for all $x\in {l}^{p}$ but $‖{L}^{n}‖↛0\text{.}$
5. Find the norms $‖L‖,‖R‖\text{.}$
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 $‖f‖=‖a‖\text{.}$
4. Prove the following facts about adjoints of bounded linear operators on Hilbert spaces.
1. ${\left(T+S\right)}^{*}={T}^{*}+{S}^{*}$
2. ${\left(TS\right)}^{*}={S}^{*}{T}^{*}$
3. ${\left(\lambda T\right)}^{*}=\stackrel{‾}{\lambda }{T}^{*}$
4. $‖{T}^{*}T‖={‖T‖}^{2}$
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 $m×n\text{.}$ Suppose the norms in ${ℂ}^{n},{ℂ}^{m}$ are both the supremum norm. Show that $‖T‖={\text{max}}_{j}\sum _{k}|{a}_{jk}|\text{.}$
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 $m×n\text{.}$ Suppose the norms in ${ℂ}^{n},{ℂ}^{m}$ are both the ${l}^{1}$ norms. Show that $‖T‖={\text{max}}_{k}\sum _{k}|{a}_{jk}|\text{.}$

## 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 ${L}^{2}\left[0,\pi \right]\text{.}$
5. Check that the eigenvectors for $L$ are the solutions to the Sturm Liouville system in the last example $y″+\lambda y=0$ on $\left[0,\pi \right]$ with the boundary conditions $y\left(0\right)=y\left(\pi \right)=0\text{.}$