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% Homework for the course "Math 554: Linear Analysis",
% Autumn quarter, 2008, Anne Greenbaum.
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{\scriptsize Math 554, Autumn 2008} \hfill
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\large
Assignment 6. Due {\bf Fri., Nov. 14}.
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\begin{tabular}{ll}
Reading: & Class Notes, pp. 33--40 \\
& Chapter 5 in Horn and Johnson.
\end{tabular}
\vspace{.3in}
% The questions!
\begin{enumerate}
\item
Let $C^1 ([a,b])$ be the space of continuous functions on $[a,b]$ whose
derivative (one-sided derivative at the endpoints) exists and is
continuous on $[a,b]$.
\begin{description}
\item[(a)]
Suppose $u \in C([a,b]) \cap C^1 ((a,b))$ and $u'$ can be extended
continuously to $[a,b]$. Show that $u \in C^1 ([a,b])$.
\item[(b)]
Show that $\| u \| = \sup_{x \in [a,b]} | u(x) | + \sup_{x \in [a,b]}
| u' (x) |$ is a norm on $C^1 ([a,b])$ which makes $C^1 ([a,b])$ into
a Banach space. [Hint for completeness: if $\{ u_n \} \subset C^1 ([a,b])$
satisfies $u_n \rightarrow u$ uniformly and $u'_n \rightarrow v$
uniformly, take limits in the equation $u_n (x) - u_n (a) =
\int_a^x u'_n (s)\,ds$ to show that $u \in C^1 ([a,b])$ and $u' = v$.]
\end{description}
\item
If $0 < \alpha \leq 1$, a function $u \in C([a,b])$ is said to satisfy
a H\"{o}lder condition of order $\alpha$ (or to be H\"{o}lder continuous
of order $\alpha$) if
\[
\sup_{x \neq y} \frac{| u(x) - u(y) |}{| x-y |^{\alpha}} < \infty .
\]
Denote by $\wedge^{\alpha} ([a,b])$ this set of functions.
\begin{description}
\item[(a)] Show that
\[
\| u \|_{\alpha} = \sup_{x \in [a,b]} | u(x) | + \sup_{x \neq y}
\frac{| u(x)-u(y) |}{| x-y |^{\alpha}}
\]
is a norm which makes $\wedge^{\alpha}$ into a Banach space.
\item[(b)] Show that $C^1 ([a,b]) \subset \wedge^{\alpha} ([a,b])$
and that the inclusion map is continuous with respect to the norms
defined above (i.e., the norm on $C^1 ([a,b])$ defined in problem 1
and that on $\wedge^{\alpha} ([a,b])$ defined in 2a).
\end{description}
\item
Prove that the dual norm to the ${\ell}^2$ norm on ${\bf F}^n$ is
again the ${\ell}^2$ norm, and show that the ${\ell}^2$
norm is the only norm on ${\bf F}^n$ with this property.
\end{enumerate}
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