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\def\classtitle{Real Analysis}
\def\classnumber{Math 70100}
\def\examtitle{Homework 4}
\def\duedate{October 4, 2019}

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

\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 {\em (Fall 2011 qual)} Let $f:\R \to \R$ be a continuous map satisfying 
$$f(x+y)=f(x)+f(y) \quad \text{for all} \quad x,y, \in \R.$$
Prove that $f$ must be linear.
\begin{sol}
\end{sol}

\question {\em (Pugh 2.37)} Let $C$ denote the vector space of continuous functions from $[0,1]$ to $\R$. This space can be endowed with the sup (or $L^\infty$) norm,
$$| f | = \sup~\{|f(x)|~:~x \in [0,1]\}$$
or the $L^1$ norm, 
$$\|f\|=\int_0^1 |f(x)|~dx.$$
Consider the identity map between $\mathit{id}$ from $(C,|\cdot|)$ to $(C,\|\cdot\|)$. 
\begin{parts}
\part Show that $\mathit{id}$ is a continuous. (Thus it is a continuous linear bijection.)
\begin{sol}
\end{sol}

\part Show that the inverse $\mathit{id}^{-1}$ is not continuous.
\begin{sol}
\end{sol}
\end{parts}

\question {\em (Folland \S 4.2 \# 14)} Let $X$ be a topological space, $Y$ a Hausdorff space, and $f$, $g$ continuous maps
from $X$ to $Y$.
\begin{parts}
\part $\{x: f(x) = g(x)\}$ is closed.
\begin{sol}
\end{sol}
\part If $f = g$ on a dense subset of $X$, then $f = g$ on all of $X$.
\begin{sol}
\end{sol}
\end{parts}

\question {\em (Lang II.5.8ab)} Let $X$ be a topological space and $E$ a vector space with norm $|\cdot|$. Let $M(X,E)$
denote the set of all maps from $X$ to $E$. Let $B(X,E)$ denote the set of bounded maps from $X$ to $E$ endowed with the sup norm defined by $\|f\|=\sup \{|f(x)|~:~x \in X\}$. Let $BC(X,E) \subset B(X,E)$ be the set of bounded continuous maps. 
\begin{parts}
\part Show that $BC(X,E)$ is closed in $B(X,E)$. 
\begin{sol}
\end{sol}
\part A {\em Banach space} is a complete normed vector space. Show that if $E$ is a Banach space, then $B(X,E)$ 
is complete.
\begin{sol}
\end{sol}
\end{parts}

\question {\em (Modified from Lang II.5.1b)} A function $f:X \to Y$ between metric spaces is {\em uniformly continuous} if for all $\epsilon>0$, there is a $\delta>0$ so that $d_X(x_1,x_2)<\delta$
implies $d_Y\big(f(x_1),f(x_2)\big)<\epsilon$ for all $x_1,x_2 \in X$. 

Let $Y$ be a complete metric space and $X$ be a metric space.
Let $A \subset X$. Let $f:A \to Y$ be uniformly continuous,
and let $\bar A \subset X$ denote the closure of $A$. Show that there exists a unique extension of $f$ to a continuous map $\bar f:\bar A \to Y$, and show that $\bar f$ is uniformly continuous. (You may assume that $X$ and $Y$ are subsets of Banach spaces if you wish, in order to write the distance function in terms of the absolute value sign.)

\begin{sol}
\end{sol}

\question {\em (Lang II.5.12)} Let $U$ be an open subset of a normed vector space. Show that $U$ is connected if and only if $U$ is path (or arcwise) connected. (Recall that if a topological space is path connected, then it is connected. See Proposition 2.7. You do not need to prove this.) ({\em Hint}: define the notion of a path-component, which is analogous to the notion of connected component.)
\begin{sol}
\end{sol}

\question {\em (Royden-Fitzpatrick \S 12.1 \# 6)} 
Let $X$ be a set and ${\mathcal T}$ be a topology on $X$. 
Let $C(X)$ denote the collection of all continuous real-valued functions on $(X,{\mathcal T})$, and let ${\mathcal W}$ denote the weak topology induced by $C(X)$. (That is ${\mathcal W}$ is the coarsest
topology on $X$ so that every $f \in C(X)$ is continuous.)
Show that if $(X, \mathcal T)$ is normal, then the two topologies are identical.
\begin{sol}
\end{sol}

\question (Fall 2016 Qual) Let $X$, $Y$, and $Z$ be topological spaces and let $z=f(x,y)$ be a mapping from $X \times Y$ to $Z$. We say $f$ {\em is continuous in $x$} if for each $y \in Y$, the function $g(x)=f(x,y)$ is continuous. We define $f$ being continuous in $y$ analogously. Prove the following statement if it is true or give a counterexample if it is false. If $f$ is continuous in $x$ and continuous in $y$, then it is continuous. ({\em Remark:} A function $f$ which is continuous in $x$ and $y$ is said to be {\em separately continuous} (in $x$ and $y$).)
\begin{sol}
\end{sol}
\end{questions}
\end{document}
