\documentclass[12pt]{article} \usepackage{amsfonts,amsmath} \def\Cx{\mathbb{C}} \def\Nl{\mathbb{N}} \def\Ra{\mathbb{Q}} \def\Rl{\mathbb{R}} \def\Ts{\mathbb{T}} \def\Ir{\mathbb{Z}} \begin{document} \centerline{\textbf{Problem Set 7}} \centerline{\textbf{Math 201A}: \textbf{Fall 2016}} \bigskip \bigskip \noindent \textbf{Problem 1.} (a) A subset $A\subset X$ of a metric space $X$ is nowhere dense in $X$ if $\bar{A}^\circ = \emptyset$ i.e., the interior of the closure is empty. Show that $A$ is nowhere dense if and only if $\bar{A}^c$ is an open dense subset of $X$. \smallskip\noindent (b) Which of the following sets are nowhere dense in $\Rl$: $A = \left\{1/n : n\in \Nl\right\}$; $B = \Ra\cap(0,1)$; $C = \text{Cantor set}$? \smallskip\noindent (c) Show that the Baire category theorem implies that a complete metric space is not a countable union of closed, nowhere dense sets. \smallskip\noindent (d) Show that a Hamel basis of an infinite-dimensional Banach space is uncountable. \bigskip\noindent \textbf{Problem 2.} Let $X = M\oplus N$ where $M$, $N$ are closed subspaces of a Banach space $X$ with $M\cap N = \{0\}$ and $M+N = X$. Define the projection $P : X\to X$ of $X$ onto $M$ along $N$ by $Px = y$ where $x=y+z$ with $y\in M$ and $z\in N$. Prove that $P$ is bounded. \textsc{Hint.} Closed graph theorem. \bigskip \noindent \textbf{Problem 3.} (a) Let $K : X\to X$ be a bounded linear operator on a Banach space $X$ with $\|K\| <1$. Show that $I-K$ has a bounded inverse given by the uniformly convergent series $(I-K)^{-1}= I + K + K^2 + K^3+\dots$. Also show that $\left\|(I-K)^{-1}\right\| \le {1}/{(1-\|K\|)}$. \smallskip\noindent (b) Suppose that $k : [0,1] \times [0,1]\to \Rl$ is a continuous function with \[ \sup\left\{|k(x,y)| : (x,y)\in [0,1]\times[0,1]\right\} < 1. \] Use (a) to show that the Fredholm integral equation of the second kind \[ u(x) - \int_0^1 k(x,y) u(y)\, dy = f(x)\qquad 0\le x \le 1, \] has a unique solution $u\in C([0,1])$ for every $f\in C([0,1])$. Express the solution $u$ as a series, and write out explicitly an approximation of $u(x)$ valid for small $k(x,y)$, up to terms of cubic order in $k$. \smallskip\noindent (c) Show that the result in (b) can also be obtained from a contraction mapping iteration $u_{n+1}=T u_n$ with $u_0 = f$ where $T: C([0,1])\to C([0,1])$ is defined by \[ (Tu)(x) = \int_0^1 k(x,y) u(y)\, dy + f(x). \] \bigskip \noindent \textbf{Problem 4.} Define the multiplication operator $\Phi : C([0,1]) \to C([0,1])$ associated with a function $\phi \in C([0,1])$ by $\Phi f = \phi \cdot f$. \smallskip\noindent (a) Equip $C([0,1])$ with the sup-norm $\|f\|_\infty = \sup_{x\in[0,1]} |f(x)|$. Show that $\|\Phi\|_\infty = \|\phi\|_\infty$. If $(\Phi_n)$ is a sequence of multiplication operators, show that $\Phi_n \to 0$ strongly in $\mathcal{B}\left(C([0,1]), \|\cdot\|_\infty\right)$ if and only if $\Phi_n \to 0$ uniformly. % in $\mathcal{B}\left(C([0,1]), \|\cdot\|_\infty\right)$. \smallskip\noindent (a) Equip $C([0,1])$ with the one-norm $\|f\|_1 = \int_0^1 |f(x)|\, dx$. Give an example of a sequence of functions $(\phi_n)$ in $C([0,1])$ with associated multiplication operators $(\Phi_n)$ such that $\Phi_n\to 0$ strongly but not uniformly in $\mathcal{B}\left(C([0,1]), \|\cdot\|_1\right)$. \bigskip \noindent \textbf{Problem 5.} Let $K : X\to X$ be a compact linear operator on an infinite-dimensional Banach space $X$. If $K$ is one-to-one, prove that the range of $K$ is not closed. \bigskip \noindent \textbf{Problem 6.} Let $1