MAST30026 Assignments
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last update: 18 March 2014
Assignment 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?

Let $X=C[0,1]$ be the set of all continuous functions
$f:[0,1]\to \mathbb{R}\text{.}$
Recall that the supremum metric on $X$ is defined by
$${d}_{\infty}(f,g)=\text{sup}\{f\left(x\right)g\left(x\right):0\le x\le 1\}$$
and the ${L}^{1}$ metric on $X$ is defined by
$${d}_{1}(f,g)={\int}_{0}^{1}f\left(x\right)g\left(x\right)\hspace{0.17em}dx\text{.}$$
Consider the sequence $\{{f}_{1},{f}_{2},{f}_{3},\dots \}$ in
$X$ where ${f}_{n}\left(x\right)=n{x}^{n}(1x)$
for $0\le x\le 1\text{.}$

Determine whether $\left\{{f}_{n}\right\}$ converges in
$(X,{d}_{1})\text{.}$

Determine whether $\left\{{f}_{n}\right\}$ converges in
$(X,{d}_{\infty})\text{.}$
(You may use any standard results about limits of real sequences.)

Let $(X,{d}_{X})$ and
$(Y,{d}_{Y})$ be metric spaces and let
$(X\times Y,d)$ be the product metric space. Show that if
$A\subseteq X$ and $B\subseteq Y,$ then
$$\stackrel{\u203e}{A}\times \stackrel{\u203e}{B}=\stackrel{\u203e}{A\times B}\text{.}$$

Let $(X,d)$ be a metric space and let $A$ be a nonempty subset of
$X\text{.}$ Recall that for each $x\in X,$
the distance from $x$ to $A$ is
$$d(x,A)=\text{inf}\{d(x,a):a\in A\}\text{.}$$

Prove that $\stackrel{\u203e}{A}=\{x\in X:d(x,A)=0\}\text{.}$

Prove that $d(x,A)d(y,A)\le d(x,y)$
for all $x,y\in X\text{.}$ [Hint: first show that
$d(x,A)\le d(x,y)+d(y,A)\text{.]}$

Deduce the function $f:X\to \mathbb{R}$ defined by
$f\left(x\right)=d(x,A)$
is continuous.

Show that if $x\notin \stackrel{\u203e}{A}$ then
$U=\{y\in X:d(y,A)<d(x,A)\}$
is an open set in $X$ such that $\stackrel{\u203e}{A}\subset U$ and
$x\notin U\text{.}$

Determine whether the following sequences of functions converge uniformly.

${f}_{n}={x}^{n{x}^{2}},\phantom{\rule{1em}{0ex}}x\in [0,1]\text{;}$

${g}_{n}={x}^{{x}^{2}/n},\phantom{\rule{1em}{0ex}}x\in [0,1]\text{.}$

${g}_{n}={x}^{{x}^{2}/n},\phantom{\rule{1em}{0ex}}x\in \mathbb{R}\text{.}$

Let $X$ be the set of all real sequences with finitely many nonzero terms with the supremum metric: if
$\text{x}=\left({x}_{i}\right)$ and
$\text{y}=\left({y}_{i}\right)$ then
$d(\text{x},\text{y})=\text{sup}\{{x}_{i}{y}_{i}:i\in \mathbb{N}\}\text{.}$
For each $n\in \mathbb{N},$ let ${\text{x}}^{n}=(1,1/2,1/3,\dots ,1/n,0,0,\dots )\text{.}$

Show that $\left\{{\text{x}}^{n}\right\}$ is a Cauchy sequence in $X\text{.}$

Show that $\left\{{\text{x}}^{n}\right\}$ does not converge to a point in $X\text{.}$
(So $X$ is not complete.)

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 $x,y\in X\text{.}$

Explain briefly why ${d}_{f}$ is a metric on $X\text{.}$

Show that $(X,{d}_{f})$ is a complete metric space if
$f\left(X\right)$ is a closed subset of $Y\text{.}$

Let
$$f\left(x\right)=\frac{2}{2+x}\phantom{\rule{1em}{0ex}}\text{for}\hspace{0.17em}x\ge 0\text{.}$$

Show that $f$ defines a contraction mapping $f:X\to X$ when
$X=[0,\infty )$ is given the usual metric.

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 $\mathbb{R}\text{.}$

Let $X$ be a complete normed vector space over $\mathbb{R}\text{.}$ (Recall a norm
$\Vert \xb7\Vert $ satisfies
$\Vert a+b\Vert \le \Vert a\Vert +\Vert b\Vert ,$
$\Vert a\Vert \ge 0$ with $\Vert a\Vert =0$
if and only if $a=0$ and $\Vert \lambda a\Vert =\left\lambda \right\Vert a\Vert \text{.}$
We then define a metric by $d(a,b)=\Vert ab\Vert \text{.)}$
A sphere in $X$ is a set
$$S(a,r)=\{x\in X:d(x,a)=\Vert xa\Vert =r\}$$
where $a\in X$ and $r>0\text{.}$

Show that each sphere in $X$ is nowhere dense.

Show that there is no sequence of spheres $\left\{{S}_{n}\right\}$ in
$X$ whose union is $X\text{.}$

Give a geometric interpretation of the result in (b) when $X={\mathbb{R}}^{2}$ with the Euclidean norm.

Show that the result of (b) does not hold in every complete metric space $X\text{.}$

Let $(X,d)$ and $(Y,d\prime )$
be metric spaces and let $f,g:X\to Y$ be continuous.

Show that the set $\{x\in X:f\left(x\right)=g\left(x\right)\}$
is a closed subset of $X\text{.}$

Show that if $f,g:X\to \mathbb{R}$ are continuous, then
$fg$ is continuous and $\{x\in X:f\left(x\right)<g\left(x\right)\}$
is open.
Assignment 2

Let $A=\{(x,y)\in {\mathbb{R}}^{2}:{x}^{2}+{y}^{2}<1\}$
and $B=\{(x,y)\in {\mathbb{R}}^{2}:{(x2)}^{2}+{y}^{2}<1\}\text{.}$
Determine whether $X=A\cup B,$
$Y=\stackrel{\u203e}{A}\cup \stackrel{\u203e}{B}$
and $Z=\stackrel{\u203e}{A}\cup B$ are connected subsets of
${\mathbb{R}}^{2}$ with the usual topology. Explain your answers briefly.

Let $X$ be a connected topological space and let $f:X\to \mathbb{R}$
be a continuous function, where $\mathbb{R}$ has the usual topology. Show that if $f$ takes only rational values, i.e.
$f\left(X\right)\subset \mathbb{Q},$ then $f$ is a constant function.

Show that $X=\{(x,y)\in {\mathbb{R}}^{2}:xy=0\}$
is not homeomorphic to $\mathbb{R}$ (with the usual topologies). [Hint: consider the effect of removing points from $X$ and
$\mathbb{R}\text{.]}$

Prove that if $X$ and $Y$ are path connected, then $X\times Y$ is also path connected.

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.

Let $X={C}^{1}[0,1],$
$Y=C[0,1]$ so that functions in $X$
are continuously differentiable and functions in $Y$ are continuous. Define norms
$\Vert f\Vert ={\text{sup}}_{t}\leftf\left(t\right)\right$
on $Y$ and ${\Vert f\Vert}_{0}=\Vert f\Vert +\Vert f\prime \Vert $
on $X,$ where $f\prime =\frac{df}{dt}\text{.}$
Let $D:X\to Y$ be the differentiation operator
$Df=\frac{df}{dt}\text{.}$

Show that $D:(X,{\Vert \xb7\Vert}_{0})\to (Y,\Vert \xb7\Vert )$
is a bounded linear operator with $\Vert D\Vert =1\text{.}$

Show that $D:(X,\Vert \xb7\Vert )\to (Y,\Vert \xb7\Vert )$
is an unbounded linear operator.
(Hint: Consider the sequence of elements ${t}^{n}$ in $X\text{).}$

Let $\{{a}_{1},{a}_{2},\dots \}$ be a
bounded sequence of complex numbers. Define an operator $T:{l}^{2}\to {l}^{2}$
by;
$$T({b}_{1},{b}_{2},\dots )=(0,{a}_{1}{b}_{1},{a}_{2}{b}_{2},\dots )\text{.}$$

Show that $T$ is a bounded linear operator and find $\Vert T\Vert \text{.}$

Compute the adjoint operator ${T}^{*}\text{.}$

Show that ${A}^{*}A\ne A{A}^{*}$
whenever $A\ne 0\text{.}$

Find the eigenvalues of ${A}^{*}\text{.}$

Let $\left[{a}_{ij}\right]$ be an infinite complex matrix,
$i,j=1,2,\dots ,$
so that ${c}_{j}=\sum _{i}\left{a}_{ij}\right$
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
$T({b}_{1},{b}_{2},\dots )=(\sum _{j}{a}_{1j}{b}_{j},\sum _{j}{a}_{2j}{b}_{j},\dots )$
is a bounded linear operator and $\Vert T\Vert \le c\text{.}$
Deduce that $\Vert T\Vert =c\text{.}$
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