%% ================================================================
\newpage
\section{Analysis results}
\label{appx:ra}

Cite whom \ldots .

\begin{thm}[Riesz representation theorem]
Let $\msH$ be a Hilbert space.  For every $\lambda \in \msH^*$ (i.e. a
continuous linear operator $\lambda:\msH\to\R$) there exists a unique
$\phi\in\msH$ such that for all $\psi\in\msH$,
$$
	\lambda(\psi)=(\phi,\psi).
$$
\end{thm}

Mnemonic:  Every row vector comes from a column vector.

\begin{thm}[Riesz-Fischer theorem]
A function is square integrable (i.e. is in $L^2$) iff its Fourier coefficients
are square-summable (i.e. are in $\ell^2$).
\begin{proof}
Sketch; needs words.

Pick (square-summable but otherwise) arbitrary $\psi \in L^2$.  [cite what]
$L^2$ has an orthonormal basis $\{\phi_k : k = 1,2, \ldots\}$.  Then
\begin{align*}
	(\psi,\psi) &= \|\psi\|^2 < \infty \\
	(\psi,\psi) &= \left(
		\sum_{j=1}^\infty c_j \phi_j,
		\sum_{k=1}^\infty c_k \phi_k \right) \\
	&= \sum_{j=1}^\infty \sum_{k=1}^\infty
		c_j \overline{c}_k \left( \phi_j, \phi_k \right) \\
	&= \sum_{j=1}^\infty \sum_{k=1}^\infty
		c_j \overline{c}_k \delta_{jk} \\
	&= \sum_{j=1}^\infty |c_j|^2 < \infty.
\end{align*}
\end{proof}
\end{thm}

%% ================================================================
\newpage
\section{Fourier expansion}
\label{appx:fourexp}

%% ----------------------------------------------------------------
\subsection{Finite dimensions}

Finite dimensional vector space $V$:  column vectors and row vectors.  Finite
sums; trivial rep theory.  Bases for $V$ and $V^*$:
$$
	\{\vece_1, \ldots, \vece_N\} \qand \{\vece_1^*, \ldots, \vece_N^*\}.
$$
For $\vecv \in V$:
$$
	\vecv = \sum_{j=1}^N a_j \vece_j.
$$
For $\lambda \in V^*$:
$$
	\lambda = \sum_{j=1}^N b_j \vece_j^*.
$$

Recall that for an orthonormal basis (namely, $(\vece_j, \vece_k) =
\delta_{jk}$), it's easy to find the $a_j$'s by the sesquilinearity of the
inner product:
\begin{align*}
	(\vecv, \vece_j)  &= \left(\sum_{j=1}^N a_k \vece_k, \vece_j \right)
	= \sum_{j=1}^N a_k \left(\vece_k, \vece_j \right)
	= \sum_{j=1}^N a_k \delta_{kj}
	= a_j.
\end{align*}

%% ----------------------------------------------------------------
\subsection{Infinite dimensions}

% what is a more-or-less standard basis for L^2(R^3)?

Separable Hilbert space $\msH$; countable basis.
(Example: $L^2([0,1])$ has basis functions
$\phi_j(x) e^{i2\pi j x}$ for $j \in \Z$.)
Cite Riesz rep thm for ``row vectors'' in Hilbert space.  For $\psi \in \msH$,
$$
	\psi = \sum_{j=1}^\infty c_j \phi_j.
$$
Cite Riesz-Fischer for
$$
	\sum_{j=1}^\infty |c_j|^2 < \infty.
$$
When $\msH=L^2(\R^3)$, equality as functions means for all $x \in \R^3$,
$$
	\psi(x) = \sum_{j=1}^\infty c_j \phi_j(x).
$$

Just as in the finite-dimensional case, it's easy to find the coefficients
if the basis is orthonormal:
\begin{align*}
	(\psi, \phi_k)  &= \left(\sum_{j=1}^N c_j \phi_j, \phi_k \right)
	= \sum_{j=1}^N c_j \left(\phi_j, \phi_k \right)
	= \sum_{j=1}^N c_j \delta_{jk}
	= c_k.
\end{align*}

For the case $\msH=L^2(\R^3)$, the inner product is represented as an
integral and we have
\begin{align*}
	(\psi,\phi_k) &= \int_{\R^3} \psi(\vecx) \phibar_k(\vecx) \,d\vecx.
\end{align*}

%% ----------------------------------------------------------------
\subsection{Tensor products of finite-dimensional vector spaces}

Do this just as a warm-up and intuition builder for the next subsection.

Oddly and only to ease the transition below, think of
$$
	V = L^2(\{1, 2, \ldots, N\}).
$$
For each index $j=1,2, \ldots, N$, we have a complex number at that index.  The
square-summability condition removes nothing, since all the sums are finite.

Then as is well-known (one hopes!) from abstract algebra, if $V$
has basis $\{\vece_j : j = 1, 2, \ldots, N\}$, then
$V \tensor V$
has basis
$$
	\{\vece_j \tensor \vece_k : j, k = 1, 2, \ldots, N\}.
$$
This basis has $N^2$ elements, and the $n$th tensor power of $V$ with
itself has dimension $N^n$.  I hope this makes it palatable that
we can think of
$$
	V\tensor V = L^2(\{1, 2, \ldots, N\}^2).
$$
and in general
$$
	V^{\tensor n} = L^2(\{1, 2, \ldots, N\}^n).
$$

For $F\in V\tensor V$,
$$
	F = \sum_{j=1}^N \sum_{k=1}^N c_{jk} \vece_j \tensor \vece_k;
$$
for $F\in V^{\tensor n}$,
$$
	F = \sum_{r_1=1}^N \cdots \sum_{r_n=1}^N
		c_{r_1, \ldots, r_n}
		\vece_{r_1} \tensor \cdots \tensor \vece_{r_n}.
$$

Something about selection of coordinates \ldots what is the appropriate notion
of inner product?  It's done by product of inner products on slots.  Write down
the formula and maybe an example; prove what is reasonable to prove.

%% ----------------------------------------------------------------
\subsection{Tensor products of infinite-dimensional vector spaces}

Tensor product of vector spaces:  If $\msH=L^2(\R^3)$, spell out the
identification of $\Htn$ with $L^2(\R^{3n})$.  Unlike in the finite-dimensional
case, the square-integrability is not for free and we have something to prove.

Certainly $\Htn$ has basis
$$
	\{\phi_{r_1} \tensor \cdots \tensor \phi_{r_n} :
		r_1, \ldots, r_n = 1, \ldots, N\}.
$$

Define
$$
	(\phi_j \tensor \phi_k)(\vecx_1, \vecx_2) :=
		\phi_j(\vecx_1) \phi_k(\vecx_2).
$$

As long as $\Htn$ is separable, it has a countable basis $\{\theta_j : j = 1,
2, \ldots\}$.  State why (use Fubini's theorem) that $\Htn$ has basis
$$
	\{\phi_{r_1}\cdot \cdots \cdot \phi_{r_n} :
		r_1, r_2, \ldots = 1, 2, \ldots\}.
$$
Prove these product functions form a complete orthonormal system (w/r/t which
inner product?).

%% ================================================================
\section{Spectra}

%% ----------------------------------------------------------------
\subsection{Finite-dimensional case}

Cite and follow \cite{bib:lax}.  Expand vector $\vecx$ and $A\vecx$ in terms of
eigenbasis.  Sum up.

%% ----------------------------------------------------------------
\subsection{Infinite-dimensional case}

Continuing to follow Lax and citing the previous section, write out the
\bfidx{spectral resolution} of the operator:  replace sum with integral.
Define \bfidx{operator-valued measure}.
List its axioms.
Cite and follow the Wikipedia article on \bfidx{projection-valued measure}.
Make some mention of \bfidx{point spectrum} and \bfidx{continuous spectrum}.
How to interpret \bfidx{singular spectrum}?
