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\begin{document}

\centerline {\bf Math 523b - Homework 4} 

\bigskip

\begin{enumerate}


\item 
\begin{enumerate}
\item Let $X,Y$ be Banach spaces. Let $\{T_n\}_{n=1}^\infty$ and $T$ 
be bounded linear operators from $X$ to $Y$. We say that $T_n$ 
converges to $T$ in the strong operator topology 
if $T_n x \ra T x$ in norm in $Y$ for all $x \in X$.
Prove that when $T_n$ converges to $T$ in the strong operator 
topology we have
$$ 
\sup_n ||T_n|| < \infty
$$
\item 
Prove that every weakly convergent sequence in $X$ is bounded. 
\end{enumerate}

\item 

\begin{enumerate}
\item Prove that a Hilbert space is separable if and only if it has 
a countable basis. {\bf You don't have to turn in this part.}
\item Let $H$ be a separable infinite dimensional Hilbert space over
$\complex$. Let $\{e_n \}_{n=1}^\infty$ be an o.n. basis. 
Let $t_{n,m} \in \complex$ for $n,m \in \naturals$. 
Suppose that 
$$
\sum_{n,m} |t_{m,n}|^2 < \infty
$$
Prove there is a bounded linear operator $T$ on $H$ such that 
$t_{m,n} = (T e_n,e_m)$. You can think of $t_{m,n}$ as the infinite 
dimensional matrix of $T$. 
(This can be proved under weaker assumptions on the $t_{n,m}$.)
\end{enumerate}


\item Problem 61, p. 178 in Folland.

\item Problem 63, p. 178 in Folland. 

\item This is a generalization of Holder's inequality. 
Let $p_1,p_2,\cdots,p_n > 0$. Let $f_i \in L^{p_i}$ for $i=1,2,\cdots,n$. 
Prove that 
$$
|| f_1 f_2 \cdots f_n ||_1 \le \prod_{i=1}^n ||f_i||_{p_i}
$$
under a suitable condition on the $p_i$. You should of course state
what the condition is.  

\item Problem 5, p. 186 in Folland. 

\bigskip

\centerline{\bf \large The following are not to be turned in }

\item Problem 57, p. 177 in Folland. 

\item Problem 7, p. 186 in Folland. 

\end{enumerate}
\end{document}
\end

