\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 8}}
\centerline{\textbf{Math 201A}: \textbf{Fall 2016}}

\bigskip
\bigskip
\noindent
\textbf{Problem 1.} (a) Let $X$ be a Banach space and $A\subset X$.
Define the annihilator $A^\perp\subset X^*$ of $A$ by
$A^\perp = \left\{F\in X^* : \text{$F(x)= 0$ for all $x\in A$}\right\}$.
Show that $A^\perp$ is a closed linear subspace of $X^*$.

\smallskip\noindent
(b) If $M$ is a linear subspace of $X$, define $J : X^*/M^\perp \to M^*$ by
\[
J\left(F + M^\perp\right) = \left.F\right|_{M},
\]
where $F\in X^*$ and $\left.F\right|_{M}$ is the restriction of $F$ to $M$.
Show that $J$ is an isometric isomoprphism. (See Problem 3 in Set 6 for quotients of Banach spaces.)


\bigskip
\noindent
\textbf{Problem 2.} (a) Let $X$ be a Banach space and $A\subset X$ a dense subset.
Suppose that $(F_n)$ is a norm-bounded sequence in the dual space $X^*$ (meaning that
$\|F_n\| \le M$ for all $n\in \Nl$) such that $F_n(x) \to f(x)$ as $n\to\infty$ for
every $x\in A$, where $f : A \to \Rl$. Prove that $(F_n)$ converges weak* to
$F\in X^*$, where $\left.F\right|_A = f$.

\smallskip\noindent
(b) Give an example of a sequence of functionals $(F_n)$ in $c_0^* = \ell^1$
and a dense subset $A\subset c_0$ such that $F_n(x) \to f(x)$ for all $x\in A$ but $(F_n)$ does
not converge weak*.


\bigskip\noindent
\textbf{Problem 3.} Let $X=\ell^\infty$ and define $F_n \in (\ell^\infty)^*$ by
$F_n(x) = x_n$ where $x = (x_k)_{k=1}^\infty$.
Show that $(F_n)$ has no weak* convergent subsequences. Why doesn't this contradict the the version
of the Banach-Alaoglu theorem we proved in class?

\bigskip\noindent
\textbf{Problem 4.} (a) Let $H$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$. Show that
the norm $\|x\| = \sqrt{\langle x,x\rangle}$ satisfies the parallelogram law
\[
2\|x\|^2 + 2\|y\|^2 = \|x+y\|^2 + \|x-y\|^2.
\]
(b) Show that the sup-norm $\|\cdot\|_\infty$ on $C([0,1])$ is not derived from an inner product.


\newpage
\noindent
\textbf{Problem 5.} Let $L^2(0,1)$ denote the Hilbert space of square-integrable functions
$f: [0,1] \to \Rl$ and define
\[
M =\left\{g\in L^2(0,1) : \int_0^1 g(x)\, dx = 0\right\}.
\]
Find the function $f\in M$ such that
\[
\int_0^1 \left|f(x) - x\right|^2\, dx = \inf_{g\in M} \int_0^1 \left|g(x) - x\right|^2\, dx.
\]


\bigskip
\noindent
\textbf{Problem 6.} Let $\ell^2(\Nl)$ denote the Hilbert space of square-summable sequences, and let
$f_n = (1,1,\dots,1,0,0,\dots)$ denote the sequence whose first $n$ terms are $1$ and whose remaining terms are zero.
Define sequences $(a_n)$, $(b_n)$, $(c_n)$ in $\ell^2(\Nl)$ by
\[
a_n = \frac{1}{n} f_n,\qquad b_n = \frac{1}{\sqrt{n}} f_n,\qquad c_n = \frac{1}{n^{1/4}} f_n.
\]
Which of these sequences converges strongly? Which of these sequences converges weakly?




\end{document}
