Last updates: 9 April 2011

- Let $X$ consist of two points $a$ and $b$,
put
$\mu \left(\right\{a\left\}\right)=\mu \left(\right\{b\left\}\right)=\frac{1}{2}$, and let
${L}^{p}\left(\mu \right)$
be the resulting
*real*${L}^{p}$-space. Identify each real function $f$ on $X$ with the point $\left(f\right(a),f(b\left)\right)$ in the plane, and sketch the unit balls of ${L}^{p}\left(\mu \right)$, for $0<p\le \infty $. 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 $\mu \left(\right\{a\left\}\right)\ne \mu \left(\right\{b\left\}\right)$, how does this situation differ from the preceding one? - Prove that the unit ball (open or closed) is convex in every normed linear space.
- If $1<p<\infty $
prove that the unit ball of ${L}^{p}\left(\mu \right)$
is
*strictly convex*; this means that if

then ${\Vert h\Vert}_{p}<1$. (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.)${\Vert f\Vert}_{p}={\Vert g\Vert}_{p}=1$, $f\ne g$, $h=\frac{1}{2}(f+g)$, - Let $C$ be the space of all continuous functions on
$[0,1]$, with the supremum norm. Let
$M$ consist of all $f\in C$ for which

Prove that $M$ is a closed convex subset of $C$ which contains no element of minimal norm.${\int}_{0}^{1/2}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt-{\int}_{1/2}^{1}=1$. - Let $M$ be the set of all $f\in {L}^{1}\left(\right[0,1\left]\right)$,
relative to Lebesgue measure, such that

Show that $M$ is a closed convex subset of ${L}^{1}\left(\right[0,1\left]\right)$ which contains infinitely many elements of minimal norm. (Compare this and Exercise 4 with Theorem 4.10.)${\int}_{0}^{1}=1$. -
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 ${M}^{\perp}$. - 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 ${L}^{1}\left(\mu \right)$.
- Let $X$ be a normed linear space, and let
${X}^{*}$ be its dual space with the norm
$\Vert f\Vert =\mathrm{sup}\left\{\phantom{\rule{0.5em}{0ex}}\left|f\right(x\left)\right|\phantom{\rule{0.5em}{0ex}}\right|\phantom{\rule{0.5em}{0ex}}\Vert x\Vert \le 1\}.$ - (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 $\Vert x\Vert $. (This gives a natural imbedding of $X$ in its "second dual" ${X}^{**}$, the dual space of ${X}^{*}$.)
- (c) Prove that $\left\{\Vert x\Vert \right\}$ is bounded if $\left\{{x}_{n}\right\}$ is a sequence in $X$ such that $\left\{f\right({x}_{n}\left)\right\}$ is bounded for every $f\in {X}^{*}$.

- 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:

${c}_{0}$ is the subspace of ${\ell}^{\infty}$ consisting of all $x\in {\ell}^{\infty}$ for which ${\xi}_{i}\to 0$ as $i\to \infty $.$x\in {\ell}^{1}$ if and only if ${\Vert x\Vert}_{1}=\sum \left|{\xi}_{i}\right|<\infty $, $x\in {\ell}^{\infty}$ if and only if ${\Vert x\Vert}_{\infty}=\mathrm{sup}\left|{\xi}_{i}\right|<\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 $\Vert \Lambda \Vert ={\Vert y\Vert}_{1}$. 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.

- 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 \left|{\alpha}_{i}\right|<\infty $.
- For $0<\alpha \le 1$,
let $\mathrm{Lip}\alpha $ denote the space
of all complex functions $f$
on $[a,b]$ for which

Prove that $\mathrm{Lip}\alpha $ is a Banach space, if $\Vert f\Vert =\left|f\right(a\left)\right|+{M}_{f}$; also if${M}_{f}=\underset{s\ne t}{\mathrm{sup}}{\displaystyle \frac{\left|f\right(s)-f(t\left)\right|}{{|s-t|}^{\alpha}}}<\infty $.

(The members of $\mathrm{Lip}\alpha $ are said to satisfy a$\Vert f\Vert ={M}_{f}+\underset{x}{\mathrm{sup}}\left|f\right(x\left)\right|$. *Lipschitz condition*of order $\alpha $.) - 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

Show that to each $({x}_{0},{y}_{0})\in K$ there corresponds a unique measure $\mu $ on $H$ such that$f(xy)=\alpha x+\beta y+\gamma $ ($\alpha ,\beta $, and $\gamma $ real).

(Compare Sec. 5.22.)$f({x}_{0},{y}_{0})={\int}_{H}f\phantom{\rule{0.2em}{0ex}}d\mu $.

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.) - 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 $\left|{f}_{n}\right(x\left)\right|<M$ for all $x\in V$ and for $n=1,2,3,\dots $.
- (b) If $\u03f5>0$, prove that there is an open set $V\ne \varnothing $ and an integer $N$ such that $\left|f\right(x)-{f}_{n}(x\left)\right|\le \u03f5$ if $x\in V$ and $n\ge N$.
*Hint for (b):*For $N=1,2,3,\dots $, put

Since $X=\bigcup {A}_{N}$, some ${A}_{N}$ has a nonempty interior.${A}_{N}=\{x\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}\left|{f}_{m}\right(x)-{f}_{n}(x\left)\right|\le \u03f5\phantom{\rule{0.5em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}m\ge N\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}n\ge N\}$.

- Let $C$ be the space of all real continuous functions on $I=[0,1]$ 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 $\left|f\right(s)-f(t\left)\right|\le n|s-t|$ 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 $\Vert g-h\Vert $ 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.
- 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

provided that these series converge.${\sigma}_{i}=\sum _{j=0}^{\infty}{a}_{ij}{s}_{j}$ ($i=1,2,3,\dots $),

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:

The process of passing from $\left\{{s}_{j}\right\}$ to $\left\{{\sigma}_{i}\right\}$ is called a$\underset{i\to \infty}{\mathrm{lim}}{a}_{ij}=0$, for each $j$. (a) ${\mathrm{sup}}_{i}\sum _{j=0}^{\infty}\left|{a}_{ij}\right|<\mathrm{\infty}$. (b) $\underset{i\to \infty}{\mathrm{lim}}\sum _{j=0}^{\infty}{a}_{ij}=1$. (c) *summability method*. Two examples are

and${a}_{ij}=\{\begin{array}{cc}\frac{1}{i+1},& \text{if}\phantom{\rule{0.5em}{0ex}}0\le j\le i,\\ 0,& \text{if}\phantom{\rule{0.5em}{0ex}}i<j,\end{array}$

Prove that each of these also transforms some divergent sequences $\left\{{s}_{j}\right\}$ (even some unbounded ones) to convergent sequences $\left\{{\sigma}_{i}\right\}$.${a}_{ij}=(1-{r}_{i}){r}_{i}^{j},\phantom{\rule{2em}{0ex}}0<{r}_{i}<1,\phantom{\rule{2em}{0ex}}{r}_{i}\to 1.$ - 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 $(x,y)$, $x\in X$ and $y\in Y$, with addition and scalar multiplication defined componentwise. Prove that $X\oplus Y$ is a Banach space, if $\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert $. The graph $G$ of $\Lambda $ is the subset of $X\oplus Y$ formed by the pairs $(x,\Lambda x)$, $x\in X$. Note that our hypothesis says that $G$ is closed; hence $G$ is a Banach space. Note that $(x,\Lambda x)\to x$ is continuous, one-to-one, and linear and maps $G$ onto $X$.

Observe that there exist*nonlinear mappings*(of ${\mathbb{R}}^{1}$ onto ${\mathbb{R}}^{1}$, for instance) whose graph is closed although they are not continuous: $f\left(x\right)=1/x$ if $x\ne 0$, $f\left(0\right)=0$. - 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 $\Vert {M}_{f}\Vert \le {\Vert f\Vert}_{\infty}$. For which measures $\mu $ is it true that $\Vert {M}_{f}\Vert ={\Vert f\Vert}_{\infty}$? For which $f\in {L}^{\infty}\left(\mu \right)$ does ${M}_{f}$ map ${L}^{2}\left(\mu \right)$ onto ${L}^{2}\left(\mu \right)$?
- Suppose $\left\{{\Lambda}_{n}\right\}$ is a sequence of bounded linear transformations from a normed linear space $X$ to a Banach space $Y$, suppose $\Vert {\Lambda}_{n}\Vert \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 $x\in X$.
- 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

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 $\left\{{s}_{n}\right(f;0)/{\lambda}_{n}\}$ is unbounded.$\underset{n\to \infty}{\mathrm{lim}}{\displaystyle \frac{{\Vert {s}_{n}\Vert}_{\infty}}{\mathrm{log}n}}=0.$ *Hint:*Apply the reasoning of Exercise 18 and that of Sec. 511, with a better estimate of ${\Vert {D}_{n}\Vert}_{1}$ than was used there. - (a) Does there exist a sequence of continuous positive functions
${f}_{n}$ on ${\mathbb{R}}^{1}$ such that
$\left\{{f}_{n}\right(x\left)\right\}$ is
unbounded if and only if $x$ is rational?

(b) Replace "rational" by "irrational" in (a) and answer the resulting question.

(c) Replace "$\left\{{f}_{n}\right(x\left)\right\}$ is unbounded" by "${f}_{n}\left(x\right)\to \infty $ as $n\to \infty $" and answer the resulting analogues of (a) and (b). - Suppose $E\subseteq {\mathbb{R}}^{1}$ 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 ${\mathbb{R}}^{1}$ onto ${\mathbb{R}}^{1}$ so that $h\left(E\right)$ does not intersect $E$?
- 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
${s}_{n}({f}_{n};0)$ and
the integrals

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 $T$.$\frac{1}{\pi}}{\int}_{-\pi}^{\pi}f\left(t\right){\displaystyle \frac{\mathrm{sin}nt}{t}}\phantom{\rule{0.5em}{0ex}}dt$

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