Integration: Exercises R Ch 5

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

Last updates: 9 April 2011

Integration: Exercises R Ch 5

1. 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?
2. Prove that the unit ball (open or closed) is convex in every normed linear space.
3. If $1<p<\infty$ prove that the unit ball of $L^p\left(\mu \right)$ is strictly convex; this means that if

   ${\Vert f\Vert }_p={\Vert g\Vert }_p=1$,      $f\ne g$,      $h=\frac{1}{2}(f+g)$,

   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.)
4. 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

   ${\int }_0^{1/2}f\left(t\right)dt-{\int }_{1/2}^1=1$.

   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 $f\in L^1\left(\right[0,1\left]\right)$, relative to Lebesgue measure, such that

   ${\int }_0^1=1$.

   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.)
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

   $\Vert f\Vert =\sup \left\{\left|f\right(x\left)\right|\right|\Vert x\Vert \le 1\}.$

   (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 $f\in X^{*}$.

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 ${\Vert x\Vert }_1=\sum \left|{\xi }_i\right|<\infty$,
   $x\in {\ell }^{\infty }$ if and only if ${\Vert x\Vert }_{\infty }=\sup \left|{\xi }_i\right|<\infty$,

   $c_0$ 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 $\Vert \Lambda \Vert ={\Vert y\Vert }_1$. 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.

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

    $M_f=\underset{s\ne t}{\sup }{\displaystyle \frac{\left|f\right(s)-f(t\left)\right|}{{|s-t|}^{\alpha }}}<\infty$.

    Prove that $\mathrm{Lip}\alpha$ is a Banach space, if $\Vert f\Vert =\left|f\right(a\left)\right|+M_f$; also if

    $\Vert f\Vert =M_f+\underset{x}{\sup }\left|f\right(x\left)\right|$.

    (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

    $f(xy)=\alpha x+\beta y+\gamma$      ($\alpha ,\beta$, and $\gamma$ real).

    Show that to each $(x_0,y_0)\in K$ there corresponds a unique measure $\mu$ on $H$ such that

    $f(x_0,y_0)={\int }_{H}fd\mu$.

    (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 $f\left(x\right)=\lim f_n$ 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 $ϵ>0$, prove that there is an open set $V\ne \varnothing$ and an integer $N$ such that $\left|f\right(x)-f_n(x\left)\right|\le ϵ$ if $x\in V$ and $n\ge N$.

    Hint for (b): For $N=1,2,3,\dots$, put

    $A_N=\{x|\left|f_m\right(x)-f_n(x\left)\right|\le ϵ\text{if}m\ge N\text{and}n\ge N\}$.

    Since $X=\bigcup A_N$, some $A_N$ has a nonempty interior.

14. Let $C$ be the space of all real continuous functions on $I=[0,1]$ 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.
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:

    $\underset{i\to \infty }{\lim }{a}_{ij}=0$,      for each $j$.	(a)
    ${\sup }_i\sum _{j=0}^{\infty }\left|a_{ij}\right|<\mathrm{\infty }$.	(b)
    $\underset{i\to \infty }{\lim }\sum _{j=0}^{\infty }{a}_{ij}=1$.	(c)

    The process of passing from $\left\{s_j\right\}$ to $\left\{{\sigma }_i\right\}$ is called a summability method. Two examples are

    $a_{ij}=\{\begin{array}{cc}\frac{1}{i+1},& \text{if}0\le j\le i,\\ 0,& \text{if}i<j,\end{array}$

    and

    $a_{ij}=(1-r_i)r_i^j,0<r_i<1,r_i\to 1.$

    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 $x=\lim x_n$ and $y=\lim \Lambda x_n$ 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 ${ℝ}^1$ 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 $\Vert M_f\Vert \le {\Vert f\Vert }_{\infty }$. 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 $L^2\left(\mu \right)$?
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 $\Vert {\Lambda }_n\Vert \le M<\infty$ 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 $x\in X$.
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 $s_n/\log n\to 0$ uniformly, as $n\to \infty$, for each $f\in C\left(T\right)$. That is, prove that

    $\underset{n\to \infty }{\lim }{\displaystyle \frac{{\Vert s_n\Vert }_{\infty }}{\log n}}=0.$

    On the other hand, if ${\lambda }_n/\log n\to 0$, 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. 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.
20. (a)  Does there exist a sequence of continuous positive functions $f_n$ on ${ℝ}^1$ 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).
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 $s_n(f_n;0)$ and the integrals

    ${\displaystyle \frac{1}{\pi }}{\int }_{-\pi }^{\pi }f\left(t\right){\displaystyle \frac{\sin nt}{t}}dt$

    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.

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