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\centerline{\textbf{Problem Set 6: Math 201B}}
\centerline{Due: Friday, February 11}

\bigskip
\bigskip\noindent
\textbf{1.} Let $X$ be a (real or complex) linear space and
$P, Q : X \to X$
projections.

\smallskip\noindent
(a) Show that $I-P$ is the projection onto $\ker P$ along $\ran P$.

\smallskip\noindent
(b) The projections $P$, $Q$ are orthogonal, written $P\perp Q$, if $PQ=QP=0$.
Show that $P + Q$ is a projection if and only if $P\perp Q$.

\smallskip\noindent
(c) If the projections $P$, $Q$ commute, show that $PQ$ is the projection onto $\ran P \cap \ran Q$ along
$\ker P + \ker Q$.

\smallskip\noindent
(d) Give an example (or examples) to show that $P+Q$ need not be a projection if $PQ=0$ but $QP\ne 0$, and 
$PQ$ need not be a projection if $P$,$Q$ do not commute.


\bigskip\noindent
\textbf{2.} Let $\mathcal{H} = L^2(\Rl)$. For any Lebesgue measurable set $A \subset \Rl$, define
\[
P_A : \mathcal{H} \to \mathcal{H}
\]
by $P_A f = \chi_A f$ where $\chi_A$ is the characteristic function
of $A$. (We define $P_\emptyset = 0$.) Show that $P_A$ is an orthogonal projection. What are its
range and kernel? Show that $P_A$, $P_B$ commute.
What is $P_A P_B$? When is $P_A \perp P_B$? What is $P_A + P_B$ in that case?



\bigskip\noindent
\textbf{3.} Suppose that $\mathcal{H}$ is a separable Hilbert space with ON basis $\{e_n : n\in \Nl\}$. Let
$M$ be the closed linear span of
\[
e_1,\quad e_3,\quad e_5,\quad e_7, \quad\dots
\]
and $N$ the closed linear span of
\[
e_1 + \frac{1}{2} e_2,\quad e_3 + \frac{1}{2^2} e_4,\quad e_5 + \frac{1}{2^3} e_6,\quad
e_7 + \frac{1}{2^3} e_8\quad
\dots.
\]
(a) Show that $M \cap N = \{0\}$. If $X = M\oplus N$, show that
\[
\overline{X} = \mathcal{H},\qquad X\ne \mathcal{H}.
\]
(Thus, $X$ is an inner-product space when equipped with the $\mathcal{H}$-inner-product.) 

\smallskip\noindent
(b) Let $P : X\to X$ be the projection of $X$ onto $M$ along $N$. Show that $P$ is unbounded.



\newpage\noindent
\textbf{4.} Let $\mathcal{H} = H^1(\Ts)$ denote the Sobolev space of $2\pi$-periodic functions
in $L^2(\Ts)$ whose weak derivative belongs to $L^2(\Ts)$
with inner product
\[
\langle u, v \rangle_{\mathcal{H}} = \int_{\Ts} \left(\bar{u} v + \bar{u}^\prime v^\prime\right)\, dx.
\]
For $f\in L^2(\Ts)$, define $F : \mathcal{H} \to \Cx$ by
\[
F(v) = \int_{\Ts} \bar{f} v\, dx.
\]
Show that $F\in \mathcal{H}^\ast$ and find the element $u\in \mathcal{H}$ such that
\[
F(v) = \langle u, v \rangle_{\mathcal{H}}.
\]
What is $\|F\|_{\mathcal{H}^\ast}$?


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