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\begin{document}
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\large



\title{Math 533, Spring 2020, HW3} 
\author{Due in class, February 17}
\maketitle

\begin{problem}
Let $\sY = C([0,1])$ and $\sX = C^1([0,1])$, both equipped with the uniform norm. Prove:
\begin{enumerate}
 \item $\sX$ is not complete.
 \item The map $d/dx: \sX \to \sY$ is closed but not bounded.
\end{enumerate}

\end{problem}


\begin{problem}
 Let $\sX$ and $\sY$ be Banach spaces, and let $\{T_n\}$ be a sequence in $L(\sX, \sY)$ such that $\lim T_n x$ exists for every $x \in \sX$.  If $Tx = \lim T_n x$, prove that $T \in L(\sX, \sY)$.
\end{problem}

\begin{problem}
Let $\sX$ be a vector space of countably infinite dimension.  Prove that there is no norm on $\sX$ with respect to which $\sX$ is complete.
\end{problem}

\begin{problem}
 Let $E_n$ be the set of all $f \in C([0,1])$ for which there exists $x_0 \in [0,1]$ such that $|f(x)-f(x_0)| \leq n|x-x_0|$ for all $x \in [0,1]$.
 \begin{enumerate}
  \item Prove that $E_n$ is nowhere dense in $C([0,1])$.
  \item Show  the set of nowhere differentiable functions is residual in $C([0,1])$.
 \end{enumerate}

\end{problem}



\begin{problem}
 Let $\sX$ be a normed vector space. Prove the following.
 \begin{enumerate}
  \item Every weakly convergent sequence in $\sX$, and every weak $*$ convergent sequence in $\sX*$, is bounded w.r.t. the norm.
  \item Every weakly compact subset of $\sX$, and every weak * compact subset of $\sX^*$, is bounded w.r.t. the norm.
  \item If $\sX$ is infinite dimensional, every nonempty weakly open set in $\sX$, and every nonempty weak * open set in $X^*$ is unbounded w.r.t. the norm.
 \end{enumerate}

\end{problem}

\begin{problem}
 If $\sX$ is a separable normed vector space, show that the weak * topology on the closed unit ball in $\sX^*$ is second countable and hence metrizable.
\end{problem}

\begin{problem}
 Show that a linear subspace of a normed vector space $\sX$ is norm closed iff it is weakly closed.
\end{problem}




\end{document}

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