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\Large Math 555\hfill Winter 2015

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{\bf Homework Set 7\hfill Due Friday, March 6}
\bigskip

\begin{enumerate}
\item
Recall that $L^p(\R^n)$ consists of equivalence classes of functions 
equal a.e.
\begin{enumerate}
\item
Show that any such equivalence class can contain at most one continuous function
(so it makes sense to say that an element of $L^p(\R^n)$ is continuous).
\item
Show that if $f\in L^\infty (\R^n)$ and $f$ is continuous, then
$$
\|f\|_\infty=\sup_{x\in\R^n}|f(x)|\ . 
$$
\end{enumerate}

\bigskip
\item
Find a continuous function $f$ on $\R$ so that $f\in L^1(\R)$ but 
$f\not\in L^\infty(\R)$.

\bigskip
\item
True or False: Give a proof, or provide a counterexample.

\noindent
Suppose $\{x_n\},\ \{y_n\}$ are sequences in a Hilbert space $\cH$, and
$x,y\in\cH$.
\begin{enumerate}
\item
If $x_n\to x$ weakly and $y_n\to y$ strongly,
then $\langle x_n,y_n\rangle \to \langle x,y\rangle$.
\item
If $x_n\to x$ weakly and $y_n\to y$ weakly,
then $\langle x_n,y_n\rangle\to \langle x,y\rangle$.
\end{enumerate}

\bigskip
\item
\begin{enumerate}
\item
Let $(V,\|\cdot\|)$ be a finite-dimensional normed vector space, let  
$C\subset V$ be closed and non-empty, and let $x\in V$.
Show that there is a closest point in $C$ to $x$. Show, however,
that the closest point need not be unique even if $C$ is convex.
\end{enumerate}

\bigskip
\item
\begin{enumerate}
\item
Let $\{p_n(x)\}$ be obtained from $\{x^n:n=0,1,2,\dots\}$ by
Gram-Schmidt
in $L^2([-1,1])$. The polynomials
$$
\left\{ \sqrt{\frac{2}{2n+1}}\ p_n\right\}
$$
are called the Legendre polynomials.
Show that $\{p_n\}$ is a complete orthonormal set.
(Hint: Use the Weierstrass Approximation Theorem.)
\item
Compute $p_0,\ p_1,\ p_2$.
\item
Use (a),(b), and Hilbert space theory to find $a,b,c\in\R$ 
for which
$$
\int_{-1}^1|x^3-(a+bx+cx^2)|^2dx
$$
is minimized.  
\end{enumerate}

\bigskip
\item
Consider {\bf The Projection Theorem} for nonempty closed subspaces of 
of a Hilbert space given on page 86 of the class notes. State and prove an analogue
of this result for nonempty closed convex cones in a Hilbert space.
(Hint: use your results from last term)

\bigskip
\item
Consider the Hilbert space $L^2(\Rn)$ and the set
\[
K:=\set{f\in L^2(\Rn)}{f(x)\ge 0\ a.e.}.
\]
\begin{enumerate}
\item Show that $K$ is a closed convex cone.
\item Show that $K$ has empty interior.
\item Given $f\in L^2(\Rn)$, what is $P_K(f)$, i.e., the projection of $f$ onto $K$.
\item Given $f\in L^2(\Rn)\setminus K$, is it possible to properly separate $f$ from $K$?
If so, provide a properly separating hyperplane, if not, provide a counter-example.
\end{enumerate}

\end{enumerate}

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