## Integration: Exercises R Ch 4

1. 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?
2. Let $\left\{{x}_{n}\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}n=1,2,\dots \right\}$ be a linearly independent set of vectors in $H$. Show that the following construction yields an orthonormal set $\left\{{u}_{n}\right\}$ such that $\left\{{x}_{1},\dots {x}_{N}\right\}$ and $\left\{{u}_{1},\dots {u}_{N}\right\}$ have the same span for all $N$.
Put ${u}_{1}={x}_{1}/‖{x}_{1}‖$. 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 separable if it contains a countable dense subset.)
3. Show that ${L}^{p}\left(T\right)$ is separable if $1\le p<\infty$, but that ${L}^{\infty }\left(T\right)$ is not separable.
4. Show that $H$ is separable if and only if $H$ contains a maximal orthonormal system which is at most countable.
5. If $M=\left\{x\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}Lx=0\right\}$, where $L$ is a continuous linear functional on $H$, prove that ${M}^{\perp }$ is a space of dimension $1$ (unless $M=H$).
6. 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
 $x=\sum _{i=1}^{\infty }{c}_{n}{u}_{n}$      $\left(\text{where}\phantom{\rule{1em}{0ex}}|{c}_{n}|\le \frac{1}{n}\right)$.
Prove that $Q$ is compact. ($Q$ is called the Hilbert cube.)
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
 $x=\sum _{i=1}^{\infty }{c}_{n}{u}_{n}$      $\left(\text{where}\phantom{\rule{1em}{0ex}}|{c}_{n}|\le {\delta }_{n}\right)$.
Prove that $S$ is compact if and only if ${\sum }_{1}^{\infty }{\delta }_{n}^{2}<\infty$.
Prove that $H$ is not locally compact.
7. 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
 $\sum _{n\in {E}_{k}}{a}_{n}^{2}>1$.
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 $\sum {b}_{n}^{2}<\infty$.
8. 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.)
9. If $A\subseteq \left[0,2\pi \right]$ and $A$ is measurable, prove that
 $\underset{n\to \infty }{\mathrm{lim}}{\int }_{A}\mathrm{cos}nx\phantom{\rule{0.5em}{0ex}}dx=\underset{n\to \infty }{\mathrm{lim}}{\int }_{A}\mathrm{sin}nx\phantom{\rule{0.5em}{0ex}}dx=0$.
10. Let ${n}_{1}<{n}_{2}<{n}_{3}<\cdots$ be positive integers, and let $E$ be the set of all $x\in \left[0,2\pi \right]$ 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 $\mathrm{sin}{n}_{k}x\to ±1/\sqrt{2}$ a.e. on $E$, by Exercise 9.
11. Find a nonempty closed set $E$ in ${L}^{2}\left(T\right)$ that contains no element of smallest norm.
12. 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
 $0<\underset{k\to \infty }{\mathrm{lim}}{k}^{-1/2}{c}_{k}<\infty$.
13. Suppose $f$ is a continuous function on ${ℝ}^{1}$ with period 1. Prove that
 $0<\underset{N\to \infty }{\mathrm{lim}}\frac{1}{N}\sum _{n=1}^{N}f\left(n\alpha \right)={\int }_{0}^{1}f\left(t\right)\phantom{\rule{0.5em}{0ex}}dt$
for every irrational real number $\alpha$. Hint: Do it first for
 $f\left(t\right)=\mathrm{exp}\left(2\pi ikt\right),\phantom{\rule{2em}{0ex}}k=0,±1,±2,\dots$.
14. Compute
 $\underset{a,b,c}{\mathrm{min}}{\int }_{-1}^{1}{|{x}^{3}-a-bx-c{x}^{2}|}^{2}\phantom{\rule{0.2em}{0ex}}dx$
and find
 $\mathrm{max}{\int }_{-1}^{1}{x}^{3}g\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$
where $g$ is subject to the restrictions
 ${\int }_{-1}^{1}g\left(x\right)\phantom{\rule{0.2em}{0ex}}dx={\int }_{-1}^{1}xg\left(x\right)\phantom{\rule{0.2em}{0ex}}dx={\int }_{-1}^{1}{x}^{2}g\left(x\right)\phantom{\rule{0.2em}{0ex}}dx=0;\phantom{\rule{2em}{0ex}}{\int }_{-1}^{1}{|g\left(x\right)|}^{2}\phantom{\rule{0.2em}{0ex}}dx=1$.
15. Compute
 $\underset{a,b,c}{\mathrm{min}}{\int }_{0}^{\infty }{|{x}^{3}-a-bx-c{x}^{2}|}^{2}{e}^{-x}\phantom{\rule{0.2em}{0ex}}dx$.
State and solve the corresponding maximum problem, as in Exercise 14.
16. If ${x}_{0}\in H$ and $M$ is a closed linear subspace of $H$, prove that
 $\mathrm{min}\left\{‖x-{x}_{0}‖\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}x\in M\right\}=\mathrm{max}\left\{|\left({x}_{0},y\right)|\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}y\in {M}^{\perp },‖y‖=1\right\}$.
17. Show that there is a continuous one-to-one mapping $\gamma$ of $\left[0,1\right]$ 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 $\left[0,1\right]$.
18. Define ${u}_{s}\left(t\right)={e}^{ist}$ for all $s\in {ℝ}^{1}$, $t\in {ℝ}^{1}$. 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
 $\left(f,g\right)=\underset{A\to \infty }{\mathrm{lim}}\frac{1}{2A}{\int }_{-A}^{A}f\left(t\right)\stackrel{‾}{g\left(t\right)}\phantom{\rule{0.2em}{0ex}}dt$
exists. Show that this inner product makes $X$ into a unitary space whose completion is a non-separable HIlbert space $H$. Show also that $\left\{{u}_{s}\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}s\in {ℝ}^{1}\right\}$ is a maximal orthonormal set in $H$.
19. Fix a positive integer $N$. Put $\omega ={e}^{2\pi i/N}$. Prove the orthogonality relations
 $\frac{1}{N}\sum _{n=1}^{N}{\omega }^{nk}=\left\{\begin{array}{cc}1,& \text{if}\phantom{\rule{0.5em}{0ex}}k=0,\\ 0,& \text{if}\phantom{\rule{0.5em}{0ex}}1\le k\le N-1,\end{array}$
and use them to derive the identities
 $\left(x,y\right)=\frac{1}{N}\sum _{n=1}^{N}{‖x+{\omega }^{n}y‖}^{2}{\omega }^{n}$
that hold in every inner product space if $N\ge 3$. Show also that
 $\left(x,y\right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{‖x+{e}^{i\theta }y‖}^{2}{e}^{i\theta }\phantom{\rule{0.2em}{0ex}}d\theta$.

## Notes and References

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

## References

[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.