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\noindent
\Large Math 554\hfill Autumn 2005

\noindent
Homework Set 3\hfill Due Friday October 21
\bigskip

\begin{flushleft}
\begin{tabular}{lp{5.5in}}
\kern-.1in{\bf Reading} & Horn \& Johnson, Chapter 5\\
              & Continue your review of linear algebra
\end{tabular}
\end{flushleft}

\begin{enumerate}
\item 
(The parallelogram law is a necessary and sufficient condition for a
norm to be induced by an inner product.)
\begin{enumerate}
\item 
Show that if a norm $\|\cdot\|$ on a vector space $V$ comes from an
inner product, then $\|\cdot\|$ satisfies the parallelogram law:
$\|x+y\|^2+\|x-y\|^2=2(\|x\|^2+\|y\|^2)$.
\item 
Let $\|\cdot\|$ be the norm induced by an inner product
$\lan\cdot,\cdot\ran$ on a vector space $V$. Show the polarization identity
\begin{enumerate}
\item 
if $\F=\R$, $\lan x,y\ran=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2)$
\item 
if $\F=\C$, $\lan
x,y\ran=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2+i\|x+iy\|^2-i\|x-iy\|^2)$
\end{enumerate}
(Remark: The polarization identity expresses an inner product in terms of
the norm that it induces. The converse of part (a) is true as well: if
$\|\cdot\|$ is a norm satisfying the parallelogram law, then it is 8induced
by an inner product. If you want a challenge, try to prove this. Start by
using the polarization identity to define $\lan x,y\ran$, and then use the
parallelogram law to show that this really defines an inner product which
induces $\|\cdot\|$.)
\item 
Is $C[a,b]$ with norm $\|u\|=\sup_{[a,b]}|u(x)|$ an inner product
space?
\item 
Show that $\ell^2$ is the only inner product space among the $\ell^p$
spaces, $1\le p\le\infty$.
\end{enumerate}

\item 
A sequence $\{v_n\}$ in an inner product space $(V,
\lan\cdot,\cdot\ran)$ is said to \index{converge weakly}\emph{converge
weakly} to $v$ if $(\forall\,w\in V)\lan v_n,w\ran\to\lan v,w\ran$, and is
said to \index{converge strongly}\emph{converge strongly} to $v$ if
$\|v_n-v\|\to 0$.
\begin{enumerate}
\item 
Show that if $\dim V<\infty$ and $v_n\to v$ weakly, then $v_n\to v$
strongly.
\item 
Show that part (a) fails for $V=\ell^2$.
\item 
Show that for any $(V,\lan\cdot,\cdot\ran)$, if $v_n\to v$ strongly,
then $v_n\to v$ weakly.
\end{enumerate}

\item
\begin{enumerate}
\item 
Show that $\|x\|_\infty=\lim_{p\to\infty}\|x\|_p$, where $\|x\|_p$
denotes the $\ell^p$-norm of $x\in\C^n$.
\item 
Show that $\|u\|_\infty=\lim_{p\to\infty}\|u\|_p$, where $\|u\|_p$
denotes the $L^p$-norm of $u\in C[a,b]$.
\end{enumerate}

\item 
Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on a vector space
$V$. Suppose that all sequences $\{v_n\}\subset V$ which satisfy
$\|v_n\|_1\to 0$ also satisfy $\|v_n\|_2\to 0$, and vice-versa. Show that
$\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent norms.

\item 
Let $C^1[a,b]$ be the space of continuous functions on $[a,b]$ whose
derivative exists at each point of $[a,b]$ (one-sided derivative at the
endpoints) and defines a continuous function on $[a,b]$.
\begin{enumerate}
\item 
Suppose $u\in C([a,b])\cap C^1((a,b))$ and $u'$ extends continuously
to $[a,b]$. Show that $u\in C^1([a,b])$.
\item 
Show that $\|u\|=\sup_{[a,b]}|u(x)|+\sup_{[a,b]}|u'(x)|$ is a norm on
$C^1[a,b]$ which makes $C^1[a,b]$ into a Banach space. (Hint for
completeness: if $u_n\in C^1[a,b]$ satisfies $u_n\to u$ uniformly and
$u'_n\to v$ uniformly, take limits in the equation
$u_n(x)-u_n(a)=\int^x_au'_n(s)ds$ to show $u\in C^1[a,b]$ and $u'=v$.)
\item Sketch how the arguments in parts (a) and (b) extend to
$C^k[a,b]:\|u\|=\sup|u|+\sup|u'|+\cdots+\sup|u^{(k)}|$ makes $C^k[a,b]$ into
a Banach space (where $k$ is an integer, $k\ge 1$).
\end{enumerate}


\item
If $0<\a\le 1$, a function $u\in C[a,b]$ is said to satisfy a H\"older
condition of order $\a$ (or to be H\"older continuous of order $\a$) if
$\sup_{x\ne y} \frac{|u(x)-u(y)|}{|x-y|^\a}<\infty$. Denote by
$\Lambda^\a([a,b])$ this set of functions.
\begin{enumerate}
\item 
Show that $\|u\|_\a=\sup|u|+\sup_{x\ne
y}\frac{|u(x)-u(y)|}{|x-y|^\a}$ is a norm which makes $\Lambda^\a([a,b])$
into a Banach space.
\item 
Show that $C^1([a,b])\subset\Lambda^\a([a,b])$, and the inclusion map
is continuous with respect to the norms defined above.
\end{enumerate}

\noindent
(Remark: A function $f\in\Lambda^\a([a,b])$ for $\a=1$ is called
Lipschitz continuous.)
\item
Consider the linear operator $T:L^2((0,1))\rightarrow L^2((0,1))$ 
defined by
$$
Tf(y) = \int_0^1 (y-x)f(x)dx.
$$
\begin{enumerate}
\item
Describe the range and null space of $T$.
\item
Derive an explicit expression for $T^T$ as a mapping from
$L^2((0,1))$ to itself.
\item
Show that $T$ is normal, i.e. $T^TT=TT^T$.
\end{enumerate}
\end{enumerate}

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