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\def\classtitle{Real Analysis}
\def\classnumber{Math 70100}
\def\examtitle{Homework 9}
\def\duedate{November 22, 2019}

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

\ifprintanswers 
\noindent
{\bf Solutions:}
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%{\bf Instructions:}
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\begin{questions}
\question{\em (Folland \S 5.1 \#7)} Let $X$ be a Banach space.
\begin{parts}
\part If $T \in L(X, X)$ and $\|I - T\| < 1$ where $I$ is the identity operator, then $T$ is
invertible; in fact, the series $\sum_{n=0}^\infty (I - T)^n$ converges in $L(X, X)$ to $T^{-1}$.

\begin{sol}
\end{sol}

\part If $T \in L(X, X)$ is invertible and $\|S - T\| < \|T^{-1}\|^{-1}$, then $S$ is invertible.
Thus the set of invertible operators is open in $L(X, X)$.

\begin{sol}
\end{sol}

\end{parts}

\question{\em (Folland \S 5.1 \#12)} Let $X$ be a normed vector space and $M$ a proper closed subspace of $X$.
\label{q:Folland 12}
\begin{parts}
\part $\|x + M\| = \inf \{\|x + y\| : y \in M\}$ is a norm on $X/M$.

\begin{sol}
\end{sol}

\part For any $\epsilon > 0$ there exists $x \in X$ such that $\|x\| = 1$ and $\|x + M\| > 1 - \epsilon$.

\begin{sol}
\end{sol}

\part The projection map $\pi(x) = x + M$ from $X$ to $X/M$ has norm $1$.

\begin{sol}
\end{sol}

\part If $X$ is complete, so is $X/M$. (Use the fact that a normed vector space is complete if and only if every absolutely convergent series converges.)

\begin{sol}
\end{sol}

\part The topology defined by the quotient norm is the quotient topology (i.e., $U \subset X/M$ is open if and only if the union of equivalence classes $x+M$ in $U$ is open in $X$). (Folland refers the reader to his Exercise 28 in \S4.2.)

\begin{sol}
\end{sol}

\end{parts}

\question{\em (Folland \S 5.1 \#13)} If $\|\cdot \|$ is a seminorm on the vector space $X$, let $M = \{x  \in X: \| x \|=0\}$. Then
$M$ is a subspace, and the map $x + M \mapsto \|x\|$ is a norm on $X/M$.

\begin{sol}
\end{sol}


\question{\em (Folland \S 5.1 \#15)} Suppose that $X$ and $Y$ are normed vector spaces and $T \in L(X, Y)$. Let $N(T)$ be the {\em null space} of $T$, i.e., $N(T) = \{x \in X: Tx = 0\}$.
\begin{parts}
\part $N(T)$ is a closed subspace of $X$.

\begin{sol}
\end{sol}

\part There is a unique $S \in L(X/N(T), Y)$ such that $T = S \circ \pi$ where $\pi : X \to X/N(T)$ is the projection (see Question \ref{q:Folland 12} above). Moreover, $\|S\|=\|T\|$.

\begin{sol}
\end{sol}

\end{parts}

\question{\em (Folland \S 5.1 \#17)} A linear functional $f$ on a normed vector space $X$ is bounded if and only if $f^{-1}(\{0\})$ is
closed. (Use Question \eqref{q:Folland 12}(b).)

\begin{sol}
\end{sol}

\question Prove that if $1<p<\infty$, then the dual of $\ell^p$ is isomorphic to $\ell^q$, where $q$ is such that $\frac{1}{p}+\frac{1}{q}=1$. 
(Check that the map sending $y \in \ell^q$ to the map $x \mapsto (x,y)$
in $(\ell^p)^\ast$ is an isomorphism. {\em First Hint:} To find the inverse, observe that given $f \in (\ell^p)^\ast$ there is an obvious corresponding $y$.
{\em Second Hint:} In class we proved that if $p \in (1,\infty)$ and $x \in \ell^p$, then there is a $y \in \ell^q$ with $\|y\|=1$ such that $(x,y)=\|x\|$.)

\begin{sol}
\end{sol}



\end{questions}
\end{document}
