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\begin{document}
\begin{center}
{\bf \large Math 702 Problem Set 10} \vs \\
{\bf due Friday 5/2/2014} \vs \vs
\end{center}

\noindent
Unless otherwise stated, $X$ is a complex Hilbert space with the inner product $\langle \cdot, \cdot \rangle$ and the induced norm $\| \cdot \|$. The null-space and range of a linear map $T:X \to X$ are denoted by $N(T)$ and $R(T)$. \vs \vs

\noindent
{\bf Problem 1.} This exercise proves a family of polarization identities in inner product spaces. \vs 
\begin{enumerate}
\item[(i)]
Let $\zeta=e^{2\pi i/n}$, where $n \geq 1$ is an integer. Show that 
$$
\frac{1}{n} \sum_{k=1}^n \zeta^{kj} = \begin{cases} 1 & \text{if} \ j=0 \\ 0 & \text{if} \ 1 \leq j \leq n-1. \end{cases} \vs
$$
\item[(ii)]
Use (i) to show that for every $x,y$ in an inner product space, 
$$
\langle x,y \rangle =\frac{1}{n} \sum_{k=1}^n  \| x + \zeta^k y \|^2 \, \zeta^k
$$ 
provided that $n \geq 3$. \vs
\item[(iii)]
Prove the following continuous version of the above identity:
$$
\langle x,y \rangle = \frac{1}{2\pi} \int_0^{2\pi} \| x + e^{it} y \|^2 \, e^{it} \, dt. \vs
$$
\end{enumerate}

\noindent
{\bf Problem 2.} Assuming $Y$ is a closed subspace of $X$, prove the following statements: \vs
\begin{enumerate}
\item[(i)]
$(Y^\perp)^\perp=Y$. \vs

\item[(ii)]
The natural map $Y^\perp \to X/Y$ given by $z \mapsto z+Y$ is an isometric isomorphism (recall that the norm on $X/Y$ is defined by $\| x+Y \| = \inf_{y \in Y} \| x+y \|$). \vs

\item[(iii)]
If $x \in X$,
$$
\min \big\{  \| x -y \| : y \in Y  \big \} = \max \big \{  | \langle x, z \rangle | : z \in Y^\perp, \| z \| =1 \big \}. \vs
$$ 
\end{enumerate}

\noindent
{\bf Problem 3.} Suppose $f$ is a non-zero bounded linear functional on $X$. Prove the following statements:\vs
\begin{enumerate}
\item[(i)]
$(\ker f)^\perp$ has dimension $1$. \vs 

\item[(ii)]
$\ker f = \ker g$ for a bounded linear functional $g$ if and only if $f=\lambda g$ for some scalar $\lambda$. \vs 
\end{enumerate}

\noindent
{\bf Problem 4.} Take advantage of the fact that $L^2[-1,1]$ is a Hilbert space to compute 
$$
\min_{a,b,c \in \CC} \int_{-1}^1 |x^3-a-bx-cx^2|^2 \, dx
$$
and 
$$
\max_g \Big| \int_{-1}^1 x^3 g(x) \, dx \Big|,
$$
where $g \in L^2[-1,1]$ is subject to the restrictions
$$
\int_{-1}^1 g(x) \, dx = \int_{-1}^1 x g(x) \, dx = \int_{-1}^1 x^2 g(x) \, dx = 0 \quad \text{and} \quad \int_{-1}^1 |g(x)|^2 \, dx = 1. \vs \vs
$$

\noindent
{\bf Problem 5.} Let $T: X \to X$ be a bounded linear map. \vs
\begin{enumerate}
\item[(i)]
Show that there is a unique bounded linear map $T^\ast: X \to X$, called the {\bit adjoint} of $T$, which satisfies 
$$
\langle T(x), y \rangle = \langle x, T^\ast(y) \rangle  
$$
for every $x,y \in X$. \vs

\item[(ii)]
Verify the relations
$$
\| T^\ast \| = \| T \| \qquad \text{and} \qquad \| T T^\ast \| = \| T^\ast T \| = \| T \|^2.
$$  

\item[(iii)] 
Show that $N(T^\ast)=R(T)^\perp$ and $N(T)=R(T^\ast)^\perp$. \vs \vs
\end{enumerate}

\noindent
{\bf Problem 6.} 
\begin{enumerate}
\item[(i)]
Let $Y$ be a closed subspace of $X$ and $P:X \to X$ be the orthogonal projection onto $Y$ (so $R(P)=Y$ and $N(P)=Y^\perp$). Show that $P=P^2=P^\ast$ and $\| P \|=1$. \vs 

\item[(ii)]
Let $P:X \to X$ be a bounded linear map such that $P=P^2=P^\ast$. Show that $R(P)$ is a closed subspace of $X$ and $P$ is the orthogonal projection onto $R(P)$.\vs \vs
\end{enumerate}

\noindent
{\bf Problem 7.} (Bonus) Suppose $T: X \to X$ is linear and  
$$
\langle T(x), y \rangle = \langle x, T(y) \rangle  
$$
for every $x,y \in X$. Show that $T$ is bounded (hence $T=T^\ast$). 



\end{document}

