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\DeclareMathOperator*{\sotstarlim}{\textsc{sot*}--lim} \DeclareMathOperator*{\sotlim}{\textsc{sot}--lim} \newcommand{\wotsum}{\textsc{wot--}\!\!\sum} \newcommand{\sotsum}{\textsc{sot--}\!\!\sum} \newcommand{\wotto}{\overset{\textsc{wot}}\longrightarrow} \newcommand{\sotto}{\overset{\textsc{sot}}\longrightarrow} \newcommand{\ra}{\rightarrow} \newcommand{\la}{\leftarrow} \newcounter{exerci}[section] \renewcommand{\theexerci}{\arabic{exerci}.} \newenvironment{Ex}{\begin{list}% {\theexerci\hfill}{\usecounter{exerci}\rightmargin=0pt\leftmargin=12pt% \labelwidth=12pt\labelsep=6pt\itemsep=2ex}}{\end{list}\medbreak} \newenvironment{parts}{\begin{enumerate}\vspace{1ex}\renewcommand{\itemsep}{1ex} \renewcommand{\labelenumi}{(\alph{enumi})}}{\end{enumerate}} \newcommand{\hint}{\textbf{Hint:} } \begin{document} \thispagestyle{empty} %%%%%%% TITLE %%%%%% \begin{center}{\bf PM822 \qquad Assignment 1 \qquad Due Tuesday, January 22}\end{center} \vskip 2em \begin{Ex} \item Let $\bA$ denote the set of all continuous functions with absolutely convergent Fourier series with the norm $\|f\|_\bA := \sum_{n \in \bZ} |\hat f(n)|$. \begin{parts} \item Prove that $\bA(\bT)$ is a Banach algebra which is isometrically isomorphic to $\ell_1(\bZ)$ with convolution. \item Identify the maximal ideal space of $\bA$.\\ \hint for $\phi\in\M_{\bA}$, show that $\lambda = \phi(z)$ determines $\phi$. \item Prove Wiener's Theorem: if $f \in \bA$ and $f(z) \ne 0$ for all $|z|=1$, then $1/f$ also has an absolutely convergent Fourier series. \end{parts} \item Let $\X$ be a Banach space. Let $\Omega$ be an open subset of $\bC$. Let $f:\Omega \to \X$ be a function which is \textit{weakly analytic}, meaning that $\phi\circ f$ is analytic for every continuous linear functional $\phi\in\X'$. Fix $z_0\in\Omega$ and a closed disk $D=\ol{b_r(z_0)} \subset \Omega$. \begin{parts} \item Prove that $f$ is bounded on $D$. \hint uniform boundedness principle. \item Use Cauchy's Theorem to express $\dfrac{\phi(f(z_0+h)) - \phi(f(z_0))}{h}- (\phi\circ f)'(z_0)$ as an integral around the boundary of $D$ for all $|h|