\begin{itemize} \item[a)] Prove, if $f,g\in \mathcal A(\mathbb T)$, then their product $fg$ also belongs to $\mathcal A(\mathbb T)$. (Hint: use the $\ell^1$ inversion theorem to write $f$ and $g$ as the sums of their Fourier series, and express the Fourier coefficients of $fg$ in terms of the coefficients of $f$ and $g$.) Thus, $\mathcal A(\mathbb T)$ is a ring. \item[b)] Prove the Fourier transform is a ring isomorphism from $\mathcal A(\mathbb T)$ {\em onto} $\ell^1(\mathbb Z)$ (where the multiplication on $\ell^1(\mathbb Z)$ is convolution). \end{itemize}
\begin{itemize} \item[a)] Draw a picture of $h_n:=\one_{[-n,n]}* \one_{[-1,1]}$ and determine its $C_0(\mathbb R)$ norm. \item[b)] Show that $h_n$ is, up to a multiplicative constant independent of $n$, the Fourier transform of the $L^1$ function \begin{equation*} f_n := \frac{\sin 2\pi x \sin 2\pi nx}{x^2}. \end{equation*} (Hint: you can compute integrals, or use the $L^1$ inversion theorem.) \item[c)] Show that $\|f_n\|_1\to \infty$ as $n\to \infty$. Conclude that the Fourier transform is not surjective. (Hint: if it were surjective… .)
\begin{equation*} E-E =\{x-y:x,y\in E\} \end{equation*} contains an interval centered at the origin. (Hint: let $-E=\{-x:x\in E\}$ consider the function $h(x)=\one_{-E}*\one_E$ and apply Proposition 25.13.)
$\mu(E)>0$, then $\mu(E)\ge \delta$. [Suggestion: Apply the closed graph theorem to the inclusion map $\imath: L^r\to L^p$. Here you may wish to use the result of Problem 24.9.] Conversely, if $\mu$ does not admit sets of arbitrarily small measure, and $1\le r\le p <\infty$, then $L^r(\mu)\subset L^p(\mu)$. As an example consider the $\ell^p$ spaces.
\begin{itemize} \item[a)] Prove, if $(h_n)$ converges to $h$ in norm, then also $(h_n)$ converges to $h$ weakly. (Hint: Cauchy-Schwarz.) \item[b)] Prove, if $H$ is infinite-dimensional, and $(e_n)$ is an orthonormal sequence in $H$, then $e_n\to 0$ weakly, but $e_n\not\to 0$ in norm. (Thus weak convergence does not imply norm convergence.) \item[c)] Prove $(h_n)$ converges to $h$ in norm if and only if $(h_n)$ converges to $h$ weakly and $\|h_n\|\to \|h\|$. \item[d)] Prove if $(h_n)$ converges to $h$ weakly, then $\|h\|\le \liminf \|h_n\|$. \end{itemize}
\begin{itemize} \item[a)] Prove there is a unique bounded operator $T^*:H\to H$ satisfying $\langle Tg,h\rangle = \langle g,T^*h\rangle$ for all $g,h\in H$, and $\|T^*\|=\|T\|$. \item[b)] Prove, if $S,T\in B(H)$, then $(aS+T)^* = \overline{a}S^*+T^*$ for all $a\in \mathbb K$, and that $T^{**}=T$. \item[c)] Prove $\|T^*T\| = \|T\|^2$. \item[d)] Prove $\text{ker}T$ is a closed subspace of $H$, $\overline{(\text{ran}T)} =(\text{ker}T^*)^\bot$ and $\text{ker}T^* = (\text{ran}T)^\bot$. \end{itemize}
\def\mm{\mathscr{M}} \def\nn{\mathscr{N}} Suppose $\xx$ is a Banach space and $\mm$ and $\nn$ are closed subspaces. Show, if for each $x\in \xx$ there exist unique $m\in\mm$ and $n\in\nn$ such that \[ x=m+n, \] then the mapping $P:\xx\to\mm$ defined by $Px=m$ is bounded. Suggestion: Consider the map $\mm \times \nn \to \xx$ defined by $(m,n)\mapsto x=m+n.$
Suppose that $\xx$ is a vector space with a countably infinite basis. (That is, there is a linearly independent set $\{x_n\}\subset \xx$ such that every vector $x\in\xx$ is expressed uniquely as a {\em finite} linear combination of the $x_n$’s.) Prove there is no norm on $\xx$ under which it is complete. (Hint: consider the finite-dimensional subspaces $\xx_n:=\text{span}\{x_1,\dots x_n\}$.)
\def\yy{\mathscr{Y}} \def\emptynorm{\|\cdot\|} Suppose $\xx$ is a finite (say $n$) dimensional vector space. Prove all norms on $\xx$ are equivalent. Suggestion: Fix a basis $e_1, \dots e_n$ for $\xx$ and define $\|\sum a_k e_k\|_1 := \sum|a_k|$. It is routine to check that $\|\cdot\|_1$ is a norm on $\xx$. Now complete the following outline. \begin{enumerate}[(i)] \item Let $\|\cdot\|$ be the given norm on $\xx$. Show there is an $M$ such that $\|x\|\le M\|x\|_1$. Conclude that the mapping $\iota:(\xx,\emptynorm_1)\to (\xx,\emptynorm)$ defined by $\iota(x)=x$ is continuous; \item Show that the unit sphere $S=\{x\in\xx : \|x\|_1 =1\}$ in $(\xx,\emptynorm_1)$ is compact in the $\emptynorm_1$ topology; \item Show that the mapping $f:S\to \mathbb R$ given by $f(x)=\|x\|$ is continuous and hence attains its infimum. Show this infimum is not $0$ and finish the proof. \end{enumerate} Prove, if $\yy$ is a normed vector space, then every linear transformation $T:\xx\to\yy$ is bounded. Suggestion: First suppose the norm on $\xx$ is the norm $\emptynorm_1$ above. |