Last updates: 10 April 2011

- If $M$ is a closed subspace of $H$, prove that $M={\left({M}^{\perp}\right)}^{\perp}$. Is there a similar true statement for spaces $M$ which are not necessarily closed?
- Let $\left\{{x}_{n}\phantom{\rule{0.5em}{0ex}}\right|\phantom{\rule{0.5em}{0ex}}n=1,2,\dots \}$
be a linearly independent set of vectors in $H$. Show that the following
construction yields an orthonormal set $\left\{{u}_{n}\right\}$
such that
$\{{x}_{1},\dots {x}_{N}\}$
and
$\{{u}_{1},\dots {u}_{N}\}$
have the same span for all $N$.

Put ${u}_{1}={x}_{1}/\Vert {x}_{1}\Vert $. Having ${u}_{1},\dots {u}_{n-1}$ define

Note that this leads to a proof of the existence of a maximal orthonormal set in separable Hilbert spaces which makes no appeal to the Hausdorff maximality principle. (A space is${v}_{n}={x}_{n}-\sum _{\mathrm{=1}}^{n-1}({x}_{n},{u}_{i}){u}_{i},{u}_{n}={v}_{n}/\Vert {v}_{n}\Vert $. *separable*if it contains a countable dense subset.) - Show that ${L}^{p}\left(T\right)$ is separable if $1\le p<\infty $, but that ${L}^{\infty}\left(T\right)$ is not separable.
- Show that $H$ is separable if and only if $H$ contains a maximal orthonormal system which is at most countable.
- If $M=\left\{x\phantom{\rule{0.5em}{0ex}}\right|\phantom{\rule{0.5em}{0ex}}Lx=0\}$, where $L$ is a continuous linear functional on $H$, prove that ${M}^{\perp}$ is a space of dimension $1$ (unless $M=H$).
- Let $\left\{{u}_{n}\right\}$ be an orthonormal set in
$H$. Show that this gives an example of a closed and bounded set which is not compact.
Let $Q$ be the set of all $x\in H$ of the
form

Prove that $Q$ is compact. ($Q$ is called the Hilbert cube.)$x=\sum _{i=1}^{\infty}{c}_{n}{u}_{n}$ $(\text{where}\phantom{\rule{1em}{0ex}}\left|{c}_{n}\right|\le \frac{1}{n})$.

More generally, let $\left\{{\delta}_{n}\right\}$ be a sequence of positive numbers, and let $S$ be the set of all $x\in H$ of the form

Prove that $S$ is compact if and only if ${\sum}_{1}^{\infty}{\delta}_{n}^{2}<\infty $.$x=\sum _{i=1}^{\infty}{c}_{n}{u}_{n}$ $(\text{where}\phantom{\rule{1em}{0ex}}\left|{c}_{n}\right|\le {\delta}_{n})$.

Prove that $H$ is not locally compact. - Suppose $\left\{{a}_{n}\right\}$
is a sequence of positive numbers such that
$\sum {a}_{n}{b}_{n}<\infty $,
whenever ${b}_{n}\le 0$ and
$\sum {b}_{n}^{2}<\infty $.
Prove that
$\sum {a}_{n}^{2}<\infty $.

*Suggestion:*If $\sum {a}_{n}^{2}=\infty $ then there are disjoint sets ${E}_{k}$ ($k=1,2,3,\dots $) so that

Define ${b}_{n}$ so that ${b}_{n}={c}_{k}{a}_{n}$ for $n\in {E}_{k}$. For suitably chosen ${c}_{k}$, $\sum {a}_{n}{b}_{n}=\infty $ although $\sum {b}_{n}^{2}<\infty $.$\sum _{n\in {E}_{k}}{a}_{n}^{2}>1$. - If ${H}_{1}$ and ${H}_{2}$ are two Hilbert spaces, prove that one of them is isomorphic to a subspace of the other. (Note that every closed subspace of a Hilbert space is a Hilbert space.)
- If $A\subseteq [0,2\pi ]$
and $A$ is measurable, prove that
$\underset{n\to \infty}{\mathrm{lim}}{\int}_{A}\mathrm{cos}nx\phantom{\rule{0.5em}{0ex}}dx=\underset{n\to \infty}{\mathrm{lim}}{\int}_{A}\mathrm{sin}nx\phantom{\rule{0.5em}{0ex}}dx=0$. - Let ${n}_{1}<{n}_{2}<{n}_{3}<\cdots $
be positive integers, and let $E$ be the set of all
$x\in [0,2\pi ]$
at which $\left\{\mathrm{sin}{n}_{k}x\right\}$
converges. Prove that $m\left(E\right)=0$.
*Hint:*$2{\mathrm{sin}}^{2}\alpha =1-\mathrm{cos}2\alpha $, so $\mathrm{sin}{n}_{k}x\to \pm 1/\sqrt{2}$ a.e. on $E$, by Exercise 9. - Find a nonempty closed set $E$ in ${L}^{2}\left(T\right)$ that contains no element of smallest norm.
- The constants ${c}_{k}$ in Sec. 4.24 were shown to be such that
${k}^{-1}{c}_{k}$
is bounded. Estimate the relevant integral more precisely and show that
$0<\underset{k\to \infty}{\mathrm{lim}}{k}^{-1/2}{c}_{k}<\infty $. - Suppose $f$ is a continuous function on
${\mathbb{R}}^{1}$ with period 1. Prove that

for every irrational real number $\alpha $.$0<\underset{N\to \infty}{\mathrm{lim}}\frac{1}{N}\sum _{n=1}^{N}f\left(n\alpha \right)={\int}_{0}^{1}f\left(t\right)\phantom{\rule{0.5em}{0ex}}dt$ *Hint:*Do it first for$f\left(t\right)=\mathrm{exp}\left(2\pi ikt\right),\phantom{\rule{2em}{0ex}}k=0,\pm 1,\pm 2,\dots $. - Compute

and find$\underset{a,b,c}{\mathrm{min}}{\int}_{-1}^{1}{|{x}^{3}-a-bx-c{x}^{2}|}^{2}\phantom{\rule{0.2em}{0ex}}dx$

where $g$ is subject to the restrictions$\mathrm{max}{\int}_{-1}^{1}{x}^{3}g\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$ ${\int}_{-1}^{1}g\left(x\right)\phantom{\rule{0.2em}{0ex}}dx={\int}_{-1}^{1}xg\left(x\right)\phantom{\rule{0.2em}{0ex}}dx={\int}_{-1}^{1}{x}^{2}g\left(x\right)\phantom{\rule{0.2em}{0ex}}dx=0;\phantom{\rule{2em}{0ex}}{\int}_{-1}^{1}{\left|g\right(x\left)\right|}^{2}\phantom{\rule{0.2em}{0ex}}dx=1$. -
Compute

State and solve the corresponding maximum problem, as in Exercise 14.$\underset{a,b,c}{\mathrm{min}}{\int}_{0}^{\infty}{|{x}^{3}-a-bx-c{x}^{2}|}^{2}{e}^{-x}\phantom{\rule{0.2em}{0ex}}dx$. -
If ${x}_{0}\in H$ and
$M$ is a closed linear subspace of $H$, prove
that
$\mathrm{min}\left\{\Vert x-{x}_{0}\Vert \phantom{\rule{0.2em}{0ex}}\right|\phantom{\rule{0.2em}{0ex}}x\in M\}=\mathrm{max}\left\{\left|\right({x}_{0},y\left)\right|\phantom{\rule{0.2em}{0ex}}\right|\phantom{\rule{0.2em}{0ex}}y\in {M}^{\perp},\Vert y\Vert =1\}$. - Show that there is a continuous one-to-one mapping $\gamma $
of $[0,1]$ into $H$
such that
$\gamma \left(b\right)-\gamma \left(a\right)$
is orthogonal to
$\gamma \left(d\right)-\gamma \left(c\right)$
whenever $0\le a\le b\le c\le d\le 1$. ($\gamma $
may be called a "curve with orthogonal increments.")
*Hint:*Take $H={L}^{2}$, and consider characteristic functions of certain subsets of $[0,1]$. - Define ${u}_{s}\left(t\right)={e}^{ist}$
for all $s\in {\mathbb{R}}^{1}$,
$t\in {\mathbb{R}}^{1}$. Let $X$
be the complex vector space consisting of all finite linear combinations of these functions
${u}_{s}$. If
$f\in X$
and
$g\in X$, show that

exists. Show that this inner product makes $X$ into a unitary space whose completion is a non-separable HIlbert space $H$. Show also that $\left\{{u}_{s}\phantom{\rule{0.2em}{0ex}}\right|\phantom{\rule{0.2em}{0ex}}s\in {\mathbb{R}}^{1}\}$ is a maximal orthonormal set in $H$.$(f,g)=\underset{A\to \infty}{\mathrm{lim}}\frac{1}{2A}{\int}_{-A}^{A}f\left(t\right)\stackrel{\u203e}{g\left(t\right)}\phantom{\rule{0.2em}{0ex}}dt$ - Fix a positive integer $N$. Put
$\omega ={e}^{2\pi i/N}$. Prove the orthogonality relations

and use them to derive the identities$\frac{1}{N}\sum _{n=1}^{N}{\omega}^{nk}=\{\begin{array}{cc}1,& \text{if}\phantom{\rule{0.5em}{0ex}}k=0,\\ 0,& \text{if}\phantom{\rule{0.5em}{0ex}}1\le k\le N-1,\end{array}$

that hold in every inner product space if $N\ge 3$. Show also that$(x,y)=\frac{1}{N}\sum _{n=1}^{N}{\Vert x+{\omega}^{n}y\Vert}^{2}{\omega}^{n}$ $(x,y)=\frac{1}{2\pi}{\int}_{-\pi}^{\pi}{\Vert x+{e}^{i\theta}y\Vert}^{2}{e}^{i\theta}\phantom{\rule{0.2em}{0ex}}d\theta $.

These exercises are taken from [Ru, Chapt. 4] for a course in "Measure Theory" at the Masters level at University of Melbourne.

[RuB]
W. Rudin,
*Principles of Mathematical Analysis*, Third edition, McGraw-Hill, 1976.
MR??????.

[Ru]
W. Rudin,
*Real and complex analysis*, Third edition, McGraw-Hill, 1987.
MR0924157.