## MAST30026 Assignments

Last update: 18 March 2014

## Assignment 1

1. Let $A$ and $B$ be bounded subsets of a metric space $\left(X,d\right)$ such that $A\cap B\ne \varnothing \text{.}$ Show that $diam(A∪B)≤diam (A)+diam(B).$ What can you say if $A$ and $B$ are disjoint?
2. Let $X=C\left[0,1\right]$ be the set of all continuous functions $f:\left[0,1\right]\to ℝ\text{.}$ Recall that the supremum metric on $X$ is defined by $d∞(f,g)=sup { |f(x)-g(x)| :0≤x≤1 }$ and the ${L}^{1}$ metric on $X$ is defined by $d1(f,g)= ∫01|f(x)-g(x)| dx.$ Consider the sequence $\left\{{f}_{1},{f}_{2},{f}_{3},\dots \right\}$ in $X$ where ${f}_{n}\left(x\right)=n{x}^{n}\left(1-x\right)$ for $0\le x\le 1\text{.}$
1. Determine whether $\left\{{f}_{n}\right\}$ converges in $\left(X,{d}_{1}\right)\text{.}$
2. Determine whether $\left\{{f}_{n}\right\}$ converges in $\left(X,{d}_{\infty }\right)\text{.}$
(You may use any standard results about limits of real sequences.)
3. Let $\left(X,{d}_{X}\right)$ and $\left(Y,{d}_{Y}\right)$ be metric spaces and let $\left(X×Y,d\right)$ be the product metric space. Show that if $A\subseteq X$ and $B\subseteq Y,$ then $A‾×B‾= A×B‾.$
4. Let $\left(X,d\right)$ 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 $d(x,A)=inf {d(x,a):a∈A}.$
1. Prove that $\stackrel{‾}{A}=\left\{x\in X:d\left(x,A\right)=0\right\}\text{.}$
2. Prove that $|d\left(x,A\right)-d\left(y,A\right)|\le d\left(x,y\right)$ for all $x,y\in X\text{.}$ [Hint: first show that $d\left(x,A\right)\le d\left(x,y\right)+d\left(y,A\right)\text{.]}$
3. Deduce the function $f:X\to ℝ$ defined by $f\left(x\right)=d\left(x,A\right)$ is continuous.
4. Show that if $x\notin \stackrel{‾}{A}$ then $U=\left\{y\in X:d\left(y,A\right) is an open set in $X$ such that $\stackrel{‾}{A}\subset U$ and $x\notin U\text{.}$
5. Determine whether the following sequences of functions converge uniformly.
1. ${f}_{n}={x}^{-n{x}^{2}},\phantom{\rule{1em}{0ex}}x\in \left[0,1\right]\text{;}$
2. ${g}_{n}={x}^{-{x}^{2}/n},\phantom{\rule{1em}{0ex}}x\in \left[0,1\right]\text{.}$
3. ${g}_{n}={x}^{-{x}^{2}/n},\phantom{\rule{1em}{0ex}}x\in ℝ\text{.}$
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 $d\left(\text{x},\text{y}\right)=\text{sup}\left\{|{x}_{i}-{y}_{i}|:i\in ℕ\right\}\text{.}$
For each $n\in ℕ,$ let ${\text{x}}^{n}=\left(1,1/2,1/3,\dots ,1/n,0,0,\dots \right)\text{.}$
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 $\left(Y,d\right)$ be a complete metric space. Let $f:X\to Y$ be an injective function and define $df(x,y)=d (f(x),f(y))$ for $x,y\in X\text{.}$
1. Explain briefly why ${d}_{f}$ is a metric on $X\text{.}$
2. Show that $\left(X,{d}_{f}\right)$ is a complete metric space if $f\left(X\right)$ is a closed subset of $Y\text{.}$
8. Let $f(x)=22+x for x≥0.$
1. Show that $f$ defines a contraction mapping $f:X\to X$ when $X=\left[0,\infty \right)$ is given the usual metric.
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 complete normed vector space over $ℝ\text{.}$ (Recall a norm $‖·‖$ satisfies $‖a+b‖\le ‖a‖+‖b‖,$ $‖a‖\ge 0$ with $‖a‖=0$ if and only if $a=0$ and $‖\lambda a‖=|\lambda |‖a‖\text{.}$ We then define a metric by $d\left(a,b\right)=‖a-b‖\text{.)}$ A sphere in $X$ is a set $S(a,r)= { x∈X:d(x,a)= ‖x-a‖=r }$ where $a\in X$ and $r>0\text{.}$
1. Show that each sphere in $X$ is nowhere dense.
2. Show that there is no sequence of spheres $\left\{{S}_{n}\right\}$ in $X$ whose union is $X\text{.}$
3. Give a geometric interpretation of the result in (b) when $X={ℝ}^{2}$ with the Euclidean norm.
4. Show that the result of (b) does not hold in every complete metric space $X\text{.}$
10. Let $\left(X,d\right)$ and $\left(Y,d\prime \right)$ be metric spaces and let $f,g:X\to Y$ be continuous.
1. Show that the set $\left\{x\in X:f\left(x\right)=g\left(x\right)\right\}$ is a closed subset of $X\text{.}$
2. Show that if $f,g:X\to ℝ$ are continuous, then $f-g$ is continuous and $\left\{x\in X:f\left(x\right) is open.

## Assignment 2

1. Let $A=\left\{\left(x,y\right)\in {ℝ}^{2}:{x}^{2}+{y}^{2}<1\right\}$ and $B=\left\{\left(x,y\right)\in {ℝ}^{2}:{\left(x-2\right)}^{2}+{y}^{2}<1\right\}\text{.}$
Determine whether $X=A\cup B,$ $Y=\stackrel{‾}{A}\cup \stackrel{‾}{B}$ and $Z=\stackrel{‾}{A}\cup B$ are connected subsets of ${ℝ}^{2}$ with the usual topology. Explain your answers briefly.
2. Let $X$ be a connected topological space and let $f:X\to ℝ$ be a continuous function, where $ℝ$ has the usual topology. Show that if $f$ takes only rational values, i.e. $f\left(X\right)\subset ℚ,$ then $f$ is a constant function.
3. Show that $X=\left\{\left(x,y\right)\in {ℝ}^{2}:xy=0\right\}$ is not homeomorphic to $ℝ$ (with the usual topologies). [Hint: consider the effect of removing points from $X$ and $ℝ\text{.]}$
4. Prove that if $X$ and $Y$ are path connected, then $X×Y$ is also path connected.
5. Show that the following hold for subsets of a topological space $X\text{;}$
• if subsets $A,B$ are path connected and $A\cap B\ne \varnothing$ then $A\cup B$ is path connected.
• Show that every point of $X$ is contained in a unique path component, which can be defined as the largest path connected subset of $X$ containing this point.
• Give examples to show that the path components need not be open or closed.
• Prove that if $X$ is locally path connected, i.e every point of $x$ is contained in an open set $U$ which is path connected, then every path component is open.
• Conclude that if $X$ is locally path connected, then the path compo- nents coincide with the connected components.
6. Let $X={C}^{1}\left[0,1\right],$ $Y=C\left[0,1\right]$ so that functions in $X$ are continuously differentiable and functions in $Y$ are continuous. Define norms $‖f‖={\text{sup}}_{t}|f\left(t\right)|$ on $Y$ and ${‖f‖}_{0}=‖f‖+‖f\prime ‖$ on $X,$ where $f\prime =\frac{df}{dt}\text{.}$ Let $D:X\to Y$ be the differentiation operator $Df=\frac{df}{dt}\text{.}$
1. Show that $D:\left(X,{‖·‖}_{0}\right)\to \left(Y,‖·‖\right)$ is a bounded linear operator with $‖D‖=1\text{.}$
2. Show that $D:\left(X,‖·‖\right)\to \left(Y,‖·‖\right)$ is an unbounded linear operator. (Hint: Consider the sequence of elements ${t}^{n}$ in $X\text{).}$
7. Let $\left\{{a}_{1},{a}_{2},\dots \right\}$ be a bounded sequence of complex numbers. Define an operator $T:{l}^{2}\to {l}^{2}$ by; $T(b1,b2,…)= (0,a1b1,a2b2,…).$
1. Show that $T$ is a bounded linear operator and find $‖T‖\text{.}$
2. Compute the adjoint operator ${T}^{*}\text{.}$
3. Show that ${A}^{*}A\ne A{A}^{*}$ whenever $A\ne 0\text{.}$
4. Find the eigenvalues of ${A}^{*}\text{.}$
8. Let $\left[{a}_{ij}\right]$ be an infinite complex matrix, $i,j=1,2,\dots ,$ so that ${c}_{j}=\sum _{i}|{a}_{ij}|$ converges for every $j$ and the sequence ${c}_{j}$ is bounded with $c={\text{sup}}_{j}{c}_{j}\text{.}$ Show that the operator $T:{l}^{1}\to {l}^{1}$ defined by $T\left({b}_{1},{b}_{2},\dots \right)=\left(\sum _{j}{a}_{1j}{b}_{j},\sum _{j}{a}_{2j}{b}_{j},\dots \right)$ is a bounded linear operator and $‖T‖\le c\text{.}$ Deduce that $‖T‖=c\text{.}$