## Integration: Exercises R Ch 5

1. Let $X$ consist of two points $a$ and $b$, put $\mu \left(\left\{a\right\}\right)=\mu \left(\left\{b\right\}\right)=\frac{1}{2}$, and let ${L}^{p}\left(\mu \right)$ be the resulting real ${L}^{p}$-space. Identify each real function $f$ on $X$ with the point $\left(f\left(a\right),f\left(b\right)\right)$ in the plane, and sketch the unit balls of ${L}^{p}\left(\mu \right)$, for $0. Note that they are convex if and only if $1\le p\le \infty$. For which $p$ is this unit ball a square? A circle? If $\mu \left(\left\{a\right\}\right)\ne \mu \left(\left\{b\right\}\right)$, how does this situation differ from the preceding one?
2. Prove that the unit ball (open or closed) is convex in every normed linear space.
3. If $1 prove that the unit ball of ${L}^{p}\left(\mu \right)$ is strictly convex; this means that if
 ${‖f‖}_{p}={‖g‖}_{p}=1$,      $f\ne g$,      $h=\frac{1}{2}\left(f+g\right)$,
then ${‖h‖}_{p}<1$. (Geometrically, the surface of the ball contains no straight lines.) Show that this fails in every ${L}^{1}\left(\mu \right)$, in every ${L}^{\infty }\left(\mu \right)$, and in every $C\left(X\right)$. (Ignore trivialities, such as spaces consisting of only one point.)
4. Let $C$ be the space of all continuous functions on $\left[0,1\right]$, with the supremum norm. Let $M$ consist of all $f\in C$ for which
 ${\int }_{0}^{1/2}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt-{\int }_{1/2}^{1}=1$.
Prove that $M$ is a closed convex subset of $C$ which contains no element of minimal norm.
5. Let $M$ be the set of all $f\in {L}^{1}\left(\left[0,1\right]\right)$, relative to Lebesgue measure, such that
 ${\int }_{0}^{1}=1$.
Show that $M$ is a closed convex subset of ${L}^{1}\left(\left[0,1\right]\right)$ which contains infinitely many elements of minimal norm. (Compare this and Exercise 4 with Theorem 4.10.)
6. Let $f$ be a bounded linear functional on a subspace $M$ of a Hilbert space $H$. Prove that $f$ has a unique norm-preserving extension to a bounded linear functional on $H$, and that this extension vanishes on ${M}^{\perp }$.
7. Construct a bounded linear functional on some subspace of ${L}^{1}\left(\mu \right)$ which has two (hence infinitely many) distinct norm-preserving linear extensions to ${L}^{1}\left(\mu \right)$.
8. Let $X$ be a normed linear space, and let ${X}^{*}$ be its dual space with the norm
 $‖f‖=\mathrm{sup}\left\{\phantom{\rule{0.5em}{0ex}}|f\left(x\right)|\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}‖x‖\le 1\right\}.$
(a)   Prove that ${X}^{*}$ is a Banach space.
(b)   Prove that the mapping $f\to f\left(x\right)$ is, for each $x\in X$, a bounded linear functional on ${X}^{*}$, of norm $‖x‖$. (This gives a natural imbedding of $X$ in its "second dual" ${X}^{**}$, the dual space of ${X}^{*}$.)
(c)   Prove that $\left\{‖x‖\right\}$ is bounded if $\left\{{x}_{n}\right\}$ is a sequence in $X$ such that $\left\{f\left({x}_{n}\right)\right\}$ is bounded for every $f\in {X}^{*}$.
9. Let ${c}_{0}$, ${\ell }^{1}$, and ${\ell }^{\infty }$ be the Banach spaces consisting of all complex sequences $x=\left\{{\xi }_{i}\right\}$, $i=1,2,\dots$, defined as follows:
 $x\in {\ell }^{1}$ if and only if ${‖x‖}_{1}=\sum |{\xi }_{i}|<\infty$, $x\in {\ell }^{\infty }$ if and only if ${‖x‖}_{\infty }=\mathrm{sup}|{\xi }_{i}|<\infty$,
${c}_{0}$ is the subspace of ${\ell }^{\infty }$ consisting of all $x\in {\ell }^{\infty }$ for which ${\xi }_{i}\to 0$ as $i\to \infty$.
Prove the following four statements.
(a)   If $y=\left\{{\eta }_{i}\right\}\in {\ell }^{1}$ and $\Lambda x=\sum {\xi }_{i}{\eta }_{i}$ for every $x\in {c}_{0}$, then $\Lambda$ is a bounded linear functional on ${c}_{0}$ and $‖\Lambda ‖={‖y‖}_{1}$. Moreover, every $\Lambda \in {\left({c}_{0}\right)}^{*}$ is obtained this way. In brief, ${\left({c}_{0}\right)}^{*}={\ell }^{1}$.
(b)   In the same sense, ${\left({\ell }_{1}\right)}^{*}={\ell }^{\infty }$.
(c)   Every $y\in {\ell }^{1}$ induces a bounded linear functional on ${\ell }^{\infty }$, as in (a). However, this does not give all of ${\left({\ell }^{\infty }\right)}^{*}$, since ${\left({\ell }^{\infty }\right)}^{*}$ contains nontrivial functionals that vanish on all of ${c}_{0}$.
(d)   ${c}_{0}$ and ${\ell }^{1}$ are separable but ${\ell }^{\infty }$ is not.
10. If $\sum {\alpha }_{i}{\eta }_{i}$ converges for every sequence $\left\{{\xi }_{i}\right\}$ such that ${\xi }_{i}\to 0$ as $i\to \infty$, prove that $\sum |{\alpha }_{i}|<\infty$.
11. For $0<\alpha \le 1$, let $\mathrm{Lip}\alpha$ denote the space of all complex functions $f$ on $\left[a,b\right]$ for which
 ${M}_{f}=\underset{s\ne t}{\mathrm{sup}}\frac{|f\left(s\right)-f\left(t\right)|}{{|s-t|}^{\alpha }}<\infty$.
Prove that $\mathrm{Lip}\alpha$ is a Banach space, if $‖f‖=|f\left(a\right)|+{M}_{f}$; also if
 $‖f‖={M}_{f}+\underset{x}{\mathrm{sup}}|f\left(x\right)|$.
(The members of $\mathrm{Lip}\alpha$ are said to satisfy a Lipschitz condition of order $\alpha$.)
12. Let $K$ be a triangle (two dimensional figure) in the plane, let $H$ be the set consisting of the vertices of $K$, and let $A$ be the set of all real functions $f$ on $K$, of the form
 $f\left(xy\right)=\alpha x+\beta y+\gamma$      ($\alpha ,\beta$, and $\gamma$ real).
Show that to each $\left({x}_{0},{y}_{0}\right)\in K$ there corresponds a unique measure $\mu$ on $H$ such that
 $f\left({x}_{0},{y}_{0}\right)={\int }_{H}f\phantom{\rule{0.2em}{0ex}}d\mu$.
(Compare Sec. 5.22.)
Replace $K$ by a square, let $H$ again be the set of its vertices, and let $A$ be as above. Show that to each point of $K$ there still corresponds a measure on $H$, with the above property, but that uniqueness is now lost.
Can you extrapolate to a more general theorem? (Think of other figures, higher dimensional spaces.)
13. Let $\left\{{f}_{n}\right\}$ be a sequence of continuous complex functions on a (nonempty) complete metric space $X$, such that $f\left(x\right)=\mathrm{lim}{f}_{n}$ exists (as a complex number) for every $x\in X$.
(a)   Prove that there is an open set $V\ne \varnothing$ and an number $M<\infty$ such that $|{f}_{n}\left(x\right)| for all $x\in V$ and for $n=1,2,3,\dots$.
(b)   If $ϵ>0$, prove that there is an open set $V\ne \varnothing$ and an integer $N$ such that $|f\left(x\right)-{f}_{n}\left(x\right)|\le ϵ$ if $x\in V$ and $n\ge N$.
Hint for (b): For $N=1,2,3,\dots$, put
 ${A}_{N}=\left\{x\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}|{f}_{m}\left(x\right)-{f}_{n}\left(x\right)|\le ϵ\phantom{\rule{0.5em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}m\ge N\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}n\ge N\right\}$.
Since $X=\bigcup {A}_{N}$, some ${A}_{N}$ has a nonempty interior.
14. Let $C$ be the space of all real continuous functions on $I=\left[0,1\right]$ with the supremum norm. Let ${X}_{n}$ be the subset of $C$ consisting of those $f$ for which there exists a $t\in I$ such that $|f\left(s\right)-f\left(t\right)|\le n|s-t|$ for all $s\in I$. Fix $n$ and prove that each open set in $C$ contains an open set which does not intersect ${X}_{n}$. (Each $f\in C$ can be uniformly approximated by a zigzag function $g$ with very large slopes, and if $‖g-h‖$ is small, $h\notin {X}_{n}$.) Show that this implies the existence of a dense ${G}_{\delta }$ in $C$ which consists entirely of nowhere differentiable functions.
15. Let $A=\left({a}_{ij}\right)$ be an infinite matrix with complex entries, where $i,j=0,1,2,\dots$. $A$ associates with each sequence $\left\{{s}_{j}\right\}$ a sequence $\left\{{\sigma }_{i}\right\}$, defined by
 ${\sigma }_{i}=\sum _{j=0}^{\infty }{a}_{ij}{s}_{j}$      ($i=1,2,3,\dots$),
provided that these series converge.
Prove that $A$ transforms every convergent sequence $\left\{{s}_{j}\right\}$ to a sequence $\left\{{\sigma }_{i}\right\}$ which converges to the same limit if and only if the following conditions are satisfied:
 $\underset{i\to \infty }{\mathrm{lim}}{a}_{ij}=0$,      for each $j$. (a) ${\mathrm{sup}}_{i}\sum _{j=0}^{\infty }|{a}_{ij}|<\infty$. (b) $\underset{i\to \infty }{\mathrm{lim}}\sum _{j=0}^{\infty }{a}_{ij}=1$. (c)
The process of passing from $\left\{{s}_{j}\right\}$ to $\left\{{\sigma }_{i}\right\}$ is called a summability method. Two examples are
 ${a}_{ij}=\left\{\begin{array}{cc}\frac{1}{i+1},& \text{if}\phantom{\rule{0.5em}{0ex}}0\le j\le i,\\ 0,& \text{if}\phantom{\rule{0.5em}{0ex}}i
and
 ${a}_{ij}=\left(1-{r}_{i}\right){r}_{i}^{j},\phantom{\rule{2em}{0ex}}0<{r}_{i}<1,\phantom{\rule{2em}{0ex}}{r}_{i}\to 1.$
Prove that each of these also transforms some divergent sequences $\left\{{s}_{j}\right\}$ (even some unbounded ones) to convergent sequences $\left\{{\sigma }_{i}\right\}$.
16. Suppose $X$ and $Y$ are Banach spaces, and suppose $\Lambda$ is a linear mapping of $X$ into $Y$, with the following property: For every sequence $\left\{{x}_{n}\right\}$ in $X$ for which $x=\mathrm{lim}{x}_{n}$ and $y=\mathrm{lim}\Lambda {x}_{n}$ exist, if is true that $y=\Lambda x$. Prove that $\Lambda$ is continuous.
This is the so called "closed graph theorem". Hint: Let $X\oplus Y$ be the set of all ordered pairs $\left(x,y\right)$, $x\in X$ and $y\in Y$, with addition and scalar multiplication defined componentwise. Prove that $X\oplus Y$ is a Banach space, if $‖\left(x,y\right)‖=‖x‖+‖y‖$. The graph $G$ of $\Lambda$ is the subset of $X\oplus Y$ formed by the pairs $\left(x,\Lambda x\right)$, $x\in X$. Note that our hypothesis says that $G$ is closed; hence $G$ is a Banach space. Note that $\left(x,\Lambda x\right)\to x$ is continuous, one-to-one, and linear and maps $G$ onto $X$.
Observe that there exist nonlinear mappings (of ${ℝ}^{1}$ onto ${ℝ}^{1}$, for instance) whose graph is closed although they are not continuous: $f\left(x\right)=1/x$ if $x\ne 0$, $f\left(0\right)=0$.
17. If $\mu$ is a ositive measure, each $f\in {L}^{\infty }\left(\mu \right)$ defines a multiplication operator ${M}_{f}$ on ${L}^{2}\left(\mu \right)$ into ${L}^{2}\left(\mu \right)$, such that ${M}_{f}\left(g\right)=fg$. Prove that $‖{M}_{f}‖\le {‖f‖}_{\infty }$. For which measures $\mu$ is it true that $‖{M}_{f}‖={‖f‖}_{\infty }$? For which $f\in {L}^{\infty }\left(\mu \right)$ does ${M}_{f}$ map ${L}^{2}\left(\mu \right)$ onto ${L}^{2}\left(\mu \right)$?
18. Suppose $\left\{{\Lambda }_{n}\right\}$ is a sequence of bounded linear transformations from a normed linear space $X$ to a Banach space $Y$, suppose $‖{\Lambda }_{n}‖\le M<\infty$ for all $n$, and suppose there is a dense set $E\subseteq X$ such that $\left\{{\Lambda }_{n}x\right\}$ converges for each $x\in E$. Prove that $\left\{{\Lambda }_{n}x\right\}$ converges for each $x\in X$.
19. If ${s}_{n}$ is the $n$th partial sum of the Fourier series of a function $f\in C\left(T\right)$, prove that ${s}_{n}/\mathrm{log}n\to 0$ uniformly, as $n\to \infty$, for each $f\in C\left(T\right)$. That is, prove that
 $\underset{n\to \infty }{\mathrm{lim}}\frac{{‖{s}_{n}‖}_{\infty }}{\mathrm{log}n}=0.$
On the other hand, if ${\lambda }_{n}/\mathrm{log}n\to 0$, prove that there exists an $f\in C\left(T\right)$ such that the sequence $\left\{{s}_{n}\left(f;0\right)/{\lambda }_{n}\right\}$ is unbounded. Hint: Apply the reasoning of Exercise 18 and that of Sec. 511, with a better estimate of ${‖{D}_{n}‖}_{1}$ than was used there.
20. (a)  Does there exist a sequence of continuous positive functions ${f}_{n}$ on ${ℝ}^{1}$ such that $\left\{{f}_{n}\left(x\right)\right\}$ is unbounded if and only if $x$ is rational?
(b)   Replace "rational" by "irrational" in (a) and answer the resulting question.
(c)   Replace "$\left\{{f}_{n}\left(x\right)\right\}$ is unbounded" by "${f}_{n}\left(x\right)\to \infty$ as $n\to \infty$" and answer the resulting analogues of (a) and (b).
21. Suppose $E\subseteq {ℝ}^{1}$ is measurable, and $m\left(E\right)=0$. Must there be a translate $E+x$ of $E$ that does not intersect $E$? Must there be a homeomorphism $h$ of ${ℝ}^{1}$ onto ${ℝ}^{1}$ so that $h\left(E\right)$ does not intersect $E$?
22. Suppose $f\in C\left(T\right)$ and $f\in \mathrm{Lip}\alpha$ for some $\alpha >0$. (See Exercise 11.) Prove that the Fourier series of $f$ converges to $f\left(x\right)$, by completing the following outline: It is enough to consider the case $x=0$, $f\left(0\right)=0$. THe difference between the partial sums ${s}_{n}\left({f}_{n};0\right)$ and the integrals
 $\frac{1}{\pi }{\int }_{-\pi }^{\pi }f\left(t\right)\frac{\mathrm{sin}nt}{t}\phantom{\rule{0.5em}{0ex}}dt$
tends to 0 as $n\to \infty$. The function $f\left(t\right)/t$ is in ${L}^{1}\left(T\right)$. Apply the Riemann-Lebesgue lemma. More careful reasoning shows that the convergence is actually uniform on $T$.

## Notes and References

These exercises are taken from [Ru, Chapt. 5] 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.