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% Edit This:
\def\classtitle{Real Analysis}
\def\classnumber{Math 70100}
\def\examtitle{Homework 10}
\def\duedate{December 6, 2019}

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

{\bf Remark:} I am giving you extra time to complete this assignment because of Thanksgiving. I will probably assign one more homework and not collect it, so it would be good to finish before the due date.

\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}

\begin{sol}
\end{sol}

\question (Folland \S 5.2 \#20) Show that if $M$ is a finite-dimensional subspace of a normed vector space $X$, there is a
closed subspace $N$ such that $M \cap N = \{0\}$ and $M + N = X$.

\begin{sol}
\end{sol}

\question (Einseidler-Ward \S 7.1.3 \# 7.11; Folland \S 5.2 \# 23) Let $Y \subset X$ be a closed subspace of a normed vector space.

\begin{parts}
\part Show that $Y^\perp = \big\{x^\ast \in X^\ast | x^\ast(Y) = {0}\big\}$ is a closed subspace.

\begin{sol}
\end{sol}

\part Show that $(X/Y)^\ast = Y^\perp$ (that is, that there is a natural isometric isomorphism
between the two).

\begin{sol}
\end{sol}

\part Show that $Y^\ast = X^\ast /Y^\perp$.

\begin{sol}
\end{sol}

\part Conclude that $Y$ is reflexive if $X$ is reflexive.

\begin{sol}
\end{sol}

\end{parts}

\question (Einseidler-Ward \S 7.1.3 \# 7.11; Folland \S 5.2 \# 25) Let $X$ be a normed vector space and suppose that the dual $X^\ast$ is separable.
Show that $X$ is also separable. In particular, if $X$ is separable but $X^\ast$ is not, then $X$ cannot
be reflexive. Find an example of a Banach space that is not reflexive for that reason.

\begin{sol}
\end{sol}

\question (Folland \S 5.3 \# 41) Let $X$ be a vector space of countably infinite dimension (that is, every element
is a finite linear combination of members of a countably infinite linearly independent
set). There is no norm on $X$ with respect to which $X$ is complete. (Use the fact that a finite dimensional subspace of a normed vector space is homeomorphic to either $\R^n$ or $\C^n$ and is therefore complete and closed. Apply the Baire Category Theorem.)

\begin{sol}
\end{sol}

\question (Spring 2015 Qual) Use the Baire Category Theorem to show that there exists a continuous function $f:[0,1] \to \R$ which is not monotone on any interval of positive length.

\begin{sol}
\end{sol}

\question (Spring 2011 Qual) Prove there is no sequence of continuous functions $f_n:\R \to \R_{>0}$ such that the sequence $\{f_n(x)\}$ is bounded for each $x \in \Q$ and unbounded for each $x \in \R \smallsetminus \Q$.
\begin{sol}
\end{sol}



\end{questions}
\end{document}
