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\begin{document}

\title{Analysis II}
\author{Homework 8}
\date{Due in class Monday, April 8, 2019}
\maketitle

\begin{enumerate}


\item Let $V$ and $W$ be normed vector spaces and $f \colon V \to W$
be continuously differentiable and homogeneous of degree $\alpha>0$,
i.e., for every $x \in V$ and $t>0$,
\[
  f(tx) = t^\alpha \, f(x) \,.
\]
Show that
\[
  f'(x)x = \alpha \, f(x) \,.
\]

\item Let $V$ be a normed vector space, $E \subset V$ open, and $f
\colon E \to \R$ differentiable on $E$.  Suppose that $f$ has a local
maximum at some point $x \in E$.  Show that, for every $\v \in V$,
\[
  D_{\v} f(x) = 0 \,.
\]
\emph{Remark:}  If $V \in \R^n$, this implies that
\[
  \nabla f(x) \equiv (\partial_1f(x), \dots, \partial_nf(x)) = 0 \,.
\]

\item Let $X$ be a metric space and $A,B \subset X$.  We say that $A$
and $B$ are \emph{separated} if
\[
  A \cap \overline B = \emptyset = \overline A \cap B \,.
\]
Moreover, $C \subset X$ is said to be \emph{connected} if it is not
the union of two non-empty separated sets.

\begin{enumerate}
\item Show that $\overline C$ is connected if $C$ is connected.
\item Give an example that the interior of a connected set may not be
connected. 
\end{enumerate}

\item Let $X$ be a metric space and $a,b \in X$.  We say that
$\gamma \colon [0,1] \to X$ is a \emph{path from $a$ to $b$} if
$\gamma$ is continuous with $f(0)=a$ and $f(1)=b$.


\begin{enumerate}
\item Let $E \subset X$ be open, fix $a \in E$, and define
\[
  \Gamma = \{ x \in E \colon \text{there exists a path from $a$ to
  $x$} \} \,.
\]
Show that $\Gamma$ is open and closed in $E$.

\emph{Note:} Here the notion of open and closed are \emph{relative} to
$E$, i.e., we consider $E$ itself as a metric space with the metric
inherited from $X$.  E.g., if $X \in \R^2$ and $E$ is the open unit
disk centered at the origin, then
\[
  F = \{ x \in E \colon x_1, x_2 \geq 0 \}
\]
is closed in $E$, even though it is clearly not closed in $X$.

\item We say that $E \subset X$ is \emph{path-connected} if for every
$a,b \in E$ there exists a path from $a$ to $b$.  Use part (a) to
argue that for open sets, the notion of connectedness and
path-connectedness is equivalent.

\end{enumerate}

\item Let $E \subset \R^n$ be open and connecte and
$f \colon E \to \R^m$ be differentiable. Show that if $f'(x) = 0$ for
every $x \in E$, then $f$ is constant in $E$.

\item Let $V$ and $W$ be normed vector spaces and let
$A \colon V \to W$ and $B \colon W \to V$ be bounded linear operators.
Furthermore, suppose that
\[
  \lVert I - BA \rVert_{L(V)} < 1 \,.
\]
\begin{enumerate}
\item Show that $F \colon L(W,V) \to L(W,V)$, defined for every $X \in
L(W,V)$ by 
\[
  FX = X + B - BAX \,,
\]
is a contraction.
\item Conclude that $F$ has a fixed point $X^*$.  State an upper bound
for the operator norm $\lVert X^* \rVert_{L(W,V)}$.

\end{enumerate}


\end{enumerate}

\end{document}
