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\begin{document}
\begin{center}
{\bf \large Math 702 Problem Set 11} \vs \\
{\bf due Friday 5/9/2014} \vs \vs
\end{center}

\noindent
Unless otherwise stated, $X$ is a complex Hilbert space with the inner product $\langle \cdot, \cdot \rangle$ and the induced norm $\| \cdot \|$.  \vs \vs

\noindent
{\bf Problem 1.} Let $(X,\mu)$ be a measure space in which there are disjoint measurable sets of finite positive measure. Show that if $p \neq 2$, the $L^p$-norm on $X$ does not satisfy the parallelogram law
$$
\| f+g \|^2 + \| f-g \|^2 = 2 \| f \|^2 + 2 \| g \|^2.
$$
Conclude that the Banach space $L^p(\mu)$ is not a Hilbert space. \vs \vs

\noindent
{\bf Problem 2.} Let $\{ x_i \}$ be a sequence of orthogonal vectors in $X$. Show that the following conditions are equivalent: \vs
\begin{enumerate}
\item[(i)]
$\sum_{i=1}^{\infty} \| x_i \|^2$ converges; \vs

\item[(ii)]
$\sum_{i=1}^{\infty} x_i$ converges in $X$; \vs 

\item[(iii)]
$\sum_{i=1}^{\infty} \langle x_i, y \rangle$ converges for every $y \in X$. \vs
\end{enumerate}
(Hint: The Pythagorean theorem is helpful here. For (iii) $\Longrightarrow$ (i), apply the uniform boundedness principle to the functionals $f_n(y)=\sum_{i=1}^n \langle y, x_i \rangle$.) \vs \vs

\noindent
{\bf Problem 3.} Let $\{ x_n \}$ and $\{ y_n \}$ be sequences in $X$. Prove the following statements: \vs
\begin{enumerate}
\item[(i)]
If $\| x_n \| \leq 1$, $\| y_n \| \leq 1$, and $\langle x_n, y_n \rangle \to 1$, then $\| x_n-y_n \| \to 0$. \vs

\item[(ii)]
If $x_n \wto x$ and $\| x_n \| \to \| x \|$, then $x_n \to x$. \vs
\end{enumerate}  
(Hint: For (ii), use Riesz's theorem according to which every element of $X^\ast$ is of the form $x \mapsto \langle x, y \rangle$ for some $y \in X$.) \vs \vs

\noindent
{\bf Problem 4.} Let $x \in X$ and $\hat{x}_{\alpha} = \langle x , u_{\alpha} \rangle$ be the Fourier coefficients of $x$ with respect to a given orthonormal basis $\{ u_{\alpha} \}$ for $X$. Prove the following statements: \vs
\begin{enumerate}
\item[(i)]
The set $\{ \alpha : \hat{x}_\alpha \neq 0 \}$ is at most countable. \vs

\item[(ii)]
If $\{ \alpha_1, \alpha_2, \alpha_3, \ldots \}$ is the set in (i),  
$$
x = \sum_n \hat{x}_{\alpha_n} u_{\alpha_n},
$$   
Thus, the ``Fourier series'' of $x$ converges to $x$ in the norm topology of $X$. \vs \vs
\end{enumerate}  

\noindent
{\bf Problem 5.} Recall that an isomorphism between two Hilbert spaces is a bijective linear map which is inner product-preserving (equivalently, norm-preserving). \vs
\begin{enumerate}
\item[(i)]
Show that $\ell^2(A)$ is isomorphic to $\ell^2(B)$ if and only if $A$ and $B$ have the same cardinality. \vs 
\item[(ii)]
Show that a Hilbert space is separable if and only if it has an orthonormal basis which is at most countable. \vs
\item[(iii)]
Conclude that every infinite-dimensional separable Hilbert space is isomorphic to $\ell^2=\ell^2(\NN)$. \vs \vs
\end{enumerate}  
 
\noindent
{\bf Problem 6.} Recall that $\TT=\RR/\ZZ$ and $L^2(\TT)$ is the space of all measurable functions on $\TT$ (identified with $1$-periodic functions on $\RR$) such that $\| f \|_2 = (\int_0^1 |f(t)|^2 \, dt)^{1/2} < \infty$. Let $0<\alpha<1$ be irrational and $f \in L^2(\TT)$ satisfy $f(t+\alpha) = f(t)$ for a.e. $t \in \TT$. Show that $f$ is constant a.e. on $\TT$. \vs 



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