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

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 $X=C[0,1]$ be the set of all continuous functions $f:[0,1]\to ℝ\text{.}$ 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 $f_n\left(x\right)=n{x}^n(1-x)$ 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 $(X,d_{\infty })\text{.}$
   (You may use any standard results about limits of real sequences.)
3. 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 $$\bar{A}\times \bar{B}=\overline{A\times B}\text{.}$$
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 $$d(x,A)=\text{inf}\{d(x,a):a\in A\}\text{.}$$
   1. Prove that $\bar{A}=\{x\in X:d(x,A)=0\}\text{.}$
   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 $U=\{y\in X:d(y,A)<d(x,A)\}$ 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. $f_n=x^{-n{x}^2},x\in [0,1]\text{;}$
   2. $g_n=x^{-x^2/n},x\in [0,1]\text{.}$
   3. $g_n=x^{-x^2/n},x\in ℝ\text{.}$
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 $d(\text{x},\text{y})=\text{sup}\{|x_i-y_i|:i\in \mathbb{N}\}\text{.}$
   For each $n\in \mathbb{N},$ let ${\text{x}}^n=(1,1/2,1/3,\dots ,1/n,0,0,\dots )\text{.}$
   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 $$f\left(x\right)=\frac{2}{2+x}\text{for}\hspace{0.17em}x\ge 0\text{.}$$
   1. Show that $f$ defines a contraction mapping $f:X\to X$ when $X=[0,\infty )$ 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 $\Vert ·\Vert$ 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 $d(a,b)=\Vert a-b\Vert \text{.)}$ A sphere in $X$ is a set $$S(a,r)=\{x\in X:d(x,a)=\Vert x-a\Vert =r\}$$ 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 $(X,d)$ and $(Y,d\prime )$ be metric spaces and let $f,g:X\to Y$ be continuous.
    1. Show that the set $\{x\in X:f\left(x\right)=g\left(x\right)\}$ is a closed subset of $X\text{.}$
    2. Show that if $f,g:X\to ℝ$ are continuous, then $f-g$ is continuous and $\{x\in X:f\left(x\right)<g\left(x\right)\}$ is open.

Assignment 2

1. Let $A=\{(x,y)\in {ℝ}^2:x^2+y^2<1\}$ and $B=\{(x,y)\in {ℝ}^2:{(x-2)}^2+y^2<1\}\text{.}$
   Determine whether $X=A\cup B,$ $Y=\bar{A}\cup \bar{B}$ and $Z=\bar{A}\cup B$ 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. $f\left(X\right)\subset \mathbb{Q},$ then $f$ is a constant function.
3. Show that $X=\{(x,y)\in {ℝ}^2:xy=0\}$ 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 $X\times Y$ 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 $X=C^1[0,1],$ $Y=C[0,1]$ so that functions in $X$ are continuously differentiable and functions in $Y$ are continuous. Define norms $\Vert f\Vert ={\text{sup}}_t\left|f\left(t\right)\right|$ 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 $Df=\frac{df}{dt}\text{.}$
   1. Show that $D:(X,{\Vert ·\Vert }_0)\to (Y,\Vert ·\Vert )$ is a bounded linear operator with $\Vert D\Vert =1\text{.}$
   2. Show that $D:(X,\Vert ·\Vert )\to (Y,\Vert ·\Vert )$ is an unbounded linear operator. (Hint: Consider the sequence of elements $t^n$ in $X\text{).}$
7. Let $\{a_1,a_2,\dots \}$ be a bounded sequence of complex numbers. Define an operator $T:l^2\to l^2$ by; $$T(b_1,b_2,\dots )=(0,a_1{b}_1,a_2{b}_2,\dots )\text{.}$$
   1. Show that $T$ is a bounded linear operator and find $\Vert T\Vert \text{.}$
   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 $c_j=\sum _i\left|a_{ij}\right|$ 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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