\documentclass[12pt]{article}
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\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 $1<p<\infty$. Define the H\"older conjugate $p'$ of $p$ by
\[
\frac{1}{p} + \frac{1}{p'} = 1.
\]
(a) Prove Young's inequality: If $a, b \ge 0$, then
\[
ab \le \frac{a^p}{p} + \frac{b^{p'}}{p'}.
\]
\textsc{Hint.} Show that $\log x$ is concave i.e., if $0\le t \le 1$, then
\[
\log\left(tx+(1-t)y\right) \ge t \log x + (1-t) \log y.
\]
(b) Let $\ell^p = \left\{ (x_n) : \text{$x_n\in \Rl$ and $\|(x_n)\|_p < \infty$}\right\}$, where
\[
\|(x_n)\|_p = \left(\sum_{n=1}^\infty |x_n|^p\right)^{1/p}.
\]
Prove H\"older's inequality: If $x=(x_n)\in \ell^p$ and $y=(y_n)\in \ell^{p'}$, then
\[
\left|\sum_{n=1}^\infty x_n y_n\right| \le \|x\|_p \|y\|_{p'}.
\]
(c) Show that $(\ell^p)^* \cong \ell^{p'}$.


\end{document}
