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% Edit This:
\def\classtitle{Real Analysis}
\def\classnumber{Math 70100}
\def\examtitle{Homework 11}
\def\duedate{never}

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\begin{document}
\noindent
{\large \bf \classnumber: \classtitle (Prof. Hooper) \hfill Due \duedate}\\
{\large \bf \examtitle}\\

\vspace{1em}

\ifprintanswers 
\noindent
{\bf Solutions:}
\else
\noindent
%{\bf Instructions:}
%Think about how to write clear and concise answers. Write the solution in your own words.
\fi

\begin{questions}

\question (Folland \S 5.5 \# 54) For any nonempty set $A$, $\ell^2 (A)$ is complete. (There is no need to reprove the case when $A$ is countable.)

\begin{sol}
\end{sol}

\question (Folland \S 5.5 \# 56) 
If $E$ is a subset of a Hilbert space $\cH$, $(E^\perp)^\perp$ is the smallest closed subspace of $\cH$ containing $E$.

\begin{sol}
\end{sol}

\question (Folland \S 5.5 \# 57)
Suppose that $\cH$ is a Hilbert space and $T \in L(\cH,\cH)$.
\begin{parts}
\part There is a unique $T^\ast \in L(\cH, \cH)$, called the adjoint of $T$, such that $\langle Tx, y \rangle = \langle x, T^\ast y\rangle$ for all $x, y \in \cH$. (Cf. Exercise 22. We have $T^\ast = V^{-1} T^\dagger V$ where $V$ is the conjugate-linear isomorphism from $\cH$ to $\cH^\ast$ in Theorem 5.25, $(V y) (x) = \langle x, y \rangle$.)

\begin{sol}
\end{sol}

\part $\|T^\ast \| = \|T\|$, $\|T^\ast T\| = \|T\|^2$, $(aS + bT)^\ast = \bar a S^\ast + \bar b T^\ast$, $(ST)^\ast = T^\ast S^\ast$,
and $T^{\ast \ast} = T$.

\begin{sol}
\end{sol}

\part Let $R$ and $N$ denote the range and nullspace. Then, $R(T)^\perp = N(T^\ast)$ and $N(T)^\perp = \overline{R(T^\ast)}$.

\begin{sol}
\end{sol}

\part $T$ is unitary if and only if $T$ is invertible and $T^{-1}=T^\ast$. 

\begin{sol}
\end{sol}

\end{parts}

\question (Folland \S 5.4 \# 47)
Suppose that $X$ and $Y$ are Banach spaces.
\begin{parts}
\part If $\{T_n\}_{n \in \N} \subset L(X,Y)$ and $T_n  \to T$ weakly (or strongly), then $\sup \{\|T_n\|:n \in \N\} < \infty$. 

\begin{sol}
\end{sol}

\part Every weakly convergent sequence in $X$, and every weak*-convergent sequence in $X^\ast$, is bounded (with respect to the norm).

\begin{sol}
\end{sol}

\end{parts}

\question (Folland \S 5.4 \#48)
Suppose that $X$ is a Banach space.
\begin{parts}
\part The norm-closed unit ball $B = \{x \in X: \|x\| \leq 1\}$ is also weakly closed. (Use Theorem 5.8d.)

\begin{sol}
\end{sol}

\part If $E \subset X$ is bounded (with respect to the norm), so is its weak closure.

\begin{sol}
\end{sol}

\part If $F \subset X^\ast$ is bounded (with respect to the norm), so is its weak* closure.

\begin{sol}
\end{sol}

\part Every weak*-Cauchy sequence in $X^\ast$ converges. (Use Exercise 38.)

\begin{sol}
\end{sol}

\end{parts}

\question (Folland \S 5.4 \# 53)
Suppose that $X$ is a Banach space and $\{T_n\}$, $\{S_n\}$ are sequences in $L(X, X)$
such that $T_n \to T$ strongly and $S_n \to S$ strongly.
\begin{parts}
\part If $\{x_n\} \subset X$ and $\|x_n - x\| \to 0$, then $\| T_n x_n - T x\| \to 0$. (Use Exercise
47 a.)

\begin{sol}
\end{sol}

\part $T_n S_n \to T S$ strongly.

\begin{sol}
\end{sol}

\end{parts}


\end{questions}
\end{document}
