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\begin{document}
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\large



\title{Math 533, Spring 2020, HW4} 
\author{Due in class, February 24}
\maketitle

Prove the following statements regarding a Hilbert space $\sX$.

\begin{problem}
 (Polarization identity) For any $x,y \in \sX$,
 \[
  4\langle x, y \rangle = \|x+y\|^2 - \|x-y\|^2 + i \|x + iy\|^2 - i\|x-iy\|^2.
 \]

\end{problem}

\begin{problem}
 If $E \subset \sX$, $(E^\perp)^\perp$ is the smallest closed subspace of $\sX$ containing $E$.
\end{problem}

\begin{problem}
 Every closed convex set $K \subset \sX$ has a unique element of minimal norm.
\end{problem}

\begin{problem}
 Let $\sX$ be an infinite-dimensional Hilbert space.
 \begin{enumerate}
  \item Every orthonormal sequence in $\sX$ converges weakly to 0.
  \item The unit sphere $S = \{x: \|x\|=1\}$ is weakly dense in the unit ball $B = \{x: \|x\| \leq 1\}$.
 \end{enumerate}

\end{problem}

\begin{problem}
 Let $A$ and $B$ be non-empty sets. Then $\ell^2(A)$ is isomorphic to $\ell^2(B)$ iff $A$ and $B$ have equal cardinality.
\end{problem}

\begin{problem}
 (The mean ergodic theorem) Let $U$ be an unitary operator on the Hilbert space $\sX$, $\sM = \{x: Ux = x\}$, $P$ the orthogonal projection onto $\sM$, and $S_n = \frac{1}{n} \sum_0^{n-1} U^j$.  Then $S_n \to P$ strongly.
\end{problem}

In the following problems $X$ is a locally compact Hausdorff space.

\begin{problem}
Let $Y$ be a closed subset of $X$, and $\mu$ a Radon measure on $Y$.  Prove that $I(f) = \int f|_Y d\mu$ is a positive linear functional on $C_c(X)$, and its induced Radon measure $\nu$ on $X$ is given by $\nu(E) = \mu(E \cap Y)$. 
\end{problem}

\begin{problem}
 Let $\mu$ be a Radon measure on $X$.  Let $N$ be the union of all open $U \subset X$ such that $\mu(U) = 0$.  Then $N$ is open, $\mu(N) = 0$, and if $V$ is open and $V \setminus N  \neq \emptyset$ then $\mu(V) > 0$.  $N^c$ is called the \emph{support} of $\mu$.
 Prove $x \in \supp(\mu)$ iff $\int f d\mu>0$ for every $f \in C_c(X, [0,1])$ s.t $f(x)>0$.  
\end{problem}


\end{document}

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