Integration: Exercises R Ch 4

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 10 April 2011

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 $\left\{x_n\right|n=1,2,\dots \}$ 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

   $v_n=x_n-\sum _{\mathrm{<mo>=</mo><mn>1</mn>}}^{n-1}(x_n,u_i)u_i, u_n=v_n/\Vert v_n\Vert$.

   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 $M=\left\{x\right|Lx=0\}$, 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

   $x=\sum _{i=1}^{\infty }{c}_n{u}_n$      $(\text{where}\left|c_n\right|\le \frac{1}{n})$.

   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

   $x=\sum _{i=1}^{\infty }{c}_n{u}_n$      $(\text{where}\left|c_n\right|\le {\delta }_n)$.

   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 $A\subseteq [0,2\pi ]$ and $A$ is measurable, prove that

   $\underset{n\to \infty }{\lim }{\int }_A\cos nxdx=\underset{n\to \infty }{\lim }{\int }_A\sin nxdx=0$.

10. 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 $\left\{\sin n_{k}x\right\}$ converges. Prove that $m\left(E\right)=0$. Hint: $2{\sin }^2\alpha =1-\cos 2\alpha$, so $\sin n_{k}x\to \pm 1/\sqrt{2}$ 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

    $0<\underset{k\to \infty }{\lim }{k}^{-1/2}{c}_k<\infty$.

13. Suppose $f$ is a continuous function on ${ℝ}^1$ with period 1. Prove that

    $0<\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{n=1}^{N}f\left(n\alpha \right)={\int }_0^{1}f\left(t\right)dt$

    for every irrational real number $\alpha$. Hint: Do it first for

    $f\left(t\right)=\exp \left(2\pi ikt\right),k=0,\pm 1,\pm 2,\dots$.

14. Compute

    $\underset{a,b,c}{\min }{\int }_{-1}^1{|x^3-a-bx-c{x}^2|}^{2}dx$

    and find

    $\max {\int }_{-1}^1{x}^{3}g\left(x\right)dx$

    where $g$ is subject to the restrictions

    ${\int }_{-1}^{1}g\left(x\right)dx={\int }_{-1}^{1}xg\left(x\right)dx={\int }_{-1}^1{x}^{2}g\left(x\right)dx=0;{\int }_{-1}^1{\left|g\right(x\left)\right|}^{2}dx=1$.

15. Compute

    $\underset{a,b,c}{\min }{\int }_0^{\infty }{|x^3-a-bx-c{x}^2|}^2{e}^{-x}dx$.

    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

    $\min \left\{\Vert x-x_0\Vert \right|x\in M\}=\max \left\{\left|\right(x_0,y\left)\right|\right|y\in M^{\perp },\Vert y\Vert =1\}$.

17. 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 $[0,1]$.
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

    $(f,g)=\underset{A\to \infty }{\lim }\frac{1}{2A}{\int }_{-A}^{A}f\left(t\right)\overline{g\left(t\right)}dt$

    exists. Show that this inner product makes $X$ into a unitary space whose completion is a non-separable HIlbert space $H$. Show also that $\left\{u_s\right|s\in {ℝ}^1\}$ is a maximal orthonormal set in $H$.
19. Fix a positive integer $N$. Put $\omega =e^{2\pi i/N}$. Prove the orthogonality relations

    $\frac{1}{N}\sum _{n=1}^N{\omega }^{nk}=\{\begin{array}{cc}1,& \text{if}k=0,\\ 0,& \text{if}1\le k\le N-1,\end{array}$

    and use them to derive the identities

    $(x,y)=\frac{1}{N}\sum _{n=1}^N{\Vert x+{\omega }^{n}y\Vert }^2{\omega }^n$

    that hold in every inner product space if $N\ge 3$. Show also that

    $(x,y)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\Vert x+e^{i\theta }y\Vert }^2{e}^{i\theta }d\theta$.

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.

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