Metric and Hilbert Spaces

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

Last updated: 4 November 2014

Assignment 2

1. Let $$\begin{array}{rcl}{\displaystyle A}& =& {\displaystyle \{(x,y)\in {ℝ}^2\hspace{0.17em}|\hspace{0.17em}{x}^2+y^2<1\}\text{and}}\\ {\displaystyle B}& =& {\displaystyle \{(x,y)\in {ℝ}^2\hspace{0.17em}|\hspace{0.17em}{(x-2)}^2+y^2<1\}\text{.}}\end{array}$$ Determine, with proof, 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.
2. Let $X$ and $Y$ be topological spaces and assume that $Y$ is Hausdorff. Let $f:X\to Y$ and $g:X\to Y$ be continuous functions.
   1. Show that the set $\{x\in X\hspace{0.17em}|\hspace{0.17em}f\left(x\right)=g\left(x\right)\}$ is a closed subset of $X\text{.}$
   2. Show that if $f:X\to ℝ$ and $g:X\to ℝ$ are continuous then $$f-g\text{is continuous.}$$
   3. Show that if $f:X\to ℝ$ and $g:X\to ℝ$ are continuous then $$\{x\in X\hspace{0.17em}|\hspace{0.17em}f\left(x\right)<g\left(x\right)\}\text{is open.}$$
3. Let $X$ be a complete normed vector space over $ℝ\text{.}$ A sphere in $X$ is a set $$S(a,r)=\{x\in X\hspace{0.17em}|\hspace{0.17em}d(x,a)=\Vert x-a\Vert =r\},\text{for}\hspace{0.17em}a\in X\hspace{0.17em}\text{and}\hspace{0.17em}r\in {ℝ}_{>0}\text{.}$$
   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 $X\text{.}$
4. Prove that if $X$ and $Y$ are path connected, then $X\times Y$ is also path connected.
5. Let $p\in {ℝ}_{>1}$ and define $q\in {ℝ}_{>1}$ by $\frac{1}{p}+\frac{1}{q}=1\text{.}$
   1. Define the normed vector space ${\ell }^p\text{.}$
   2. Show that ${\ell }^p$ is a Banach space.
   3. Prove that the dual of ${\ell }^p$ is ${\ell }^q\text{.}$
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. $$\begin{array}{rcl}{\displaystyle Y}& =& {\displaystyle C[0,1],\text{with norm given by}\Vert f\Vert =\text{sup}\left\{\mid f\left(t\right)\mid \hspace{0.17em}\right|\hspace{0.17em}t\in [0,1]\},\hspace{0.17em}\text{and}}\\ {\displaystyle X}& =& {\displaystyle C^1[0,1],\text{with norm given by}{\Vert f\Vert }_0=\Vert f\Vert +\Vert f\prime \Vert ,}\end{array}$$ 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 if $T\ne 0$ then $T^{*}T\ne T{T}^{*}\text{.}$
   4. Find the eigenvalues of $T^{*}\text{.}$
8. Let $\left[a_{ij}\right]$ be an infinite complex matrix, $i,j=1,2,\dots ,$ such that if $j\in {\mathbb{Z}}_{>0}$ then $$c_j=\sum _i\mid a_{ij}\mid \text{converges,}\text{and}c=\text{sup}\{c_1,c_2,\dots \}<\infty \text{.}$$ Show that the operator $T:{\ell }^1\to {\ell }^1$ 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 that $\Vert T\Vert =c\text{.}$

Notes and References

These are a typed copy of Assignment 2 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces.

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