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\begin{document}

\centerline {\bf Math 523b - Final - Spring 2013}

\bigskip

\noindent 
{\bf Do four of the following five problems.} Do not turn in five
problems. If you turn in five problems, I will use your four lowest
scores to compute your total.

\medskip

\noindent 
{\bf WRITE YOUR NAME ON EACH SHEET YOU TURN IN AND DO NOT WRITE
ON THE BACK}

\bigskip

\noindent We use $dx$ to denote Lebesgue measure.

\bigskip

\begin{enumerate}

\item Let $X$ and $Y$ be locally compact topological spaces. Prove
that $X \times Y$ with the product topology is locally compact.

\item Let $1 < p < \infty$. Suppose $f_n \in L^p([0,1],dx)$ with 
$||f_n||_p \le 1$ for all $n$. Suppose also that for all $\alpha \ge 0$ 
\beann
\lim_{n \ra \infty} \int_0^1 f_n(x) e^{-\alpha x} \, dx =0
\eeann
Prove that $f_n$ converges to $0$ weakly in $L^p$. 
(You may assume all functions are real valued.)

\item Let $M$ be a closed subspace of a Hilbert space $H$.
Let $F$ be a bounded linear functional on $M$. The Hahn Banach
theorem says that there is an extension of $F$ to a linear 
functional on all of $H$ which has the same norm. Prove
that such an extension is unique.

\item
Let $C_0=C_0(\reals)$ be the real-valued 
continuous functions on the real line that 
go to zero at $\pm \infty$. We use the usual sup norm.
If $g \in L^1(\reals,dx)$ is a real-valued Lebesgue integrable 
function, then we can define a bounded linear function $\phi_g$
on $C_0$ by 
\beann 
\phi_g(f)= \int f(x) g(x) \, dx
\eeann
Let $F$ be the set of all such linear functionals:
\beann
F= \{\phi_g: g \in L^1(\reals,dx)\}
\eeann
Prove that $F$ is a closed subset of $C_0^*$.

\newpage
 
\item Let $X_n$ be an independent, identically distributed 
sequence of random variables with $X_1 \ge 0$ a.e. and 
$E[X_1]=\infty$. This implies that for all $c>0$,
\beann
\sum_{n=1}^\infty P(X_1 \ge c n) = \infty
\eeann
You may assume this fact without proving it. 

(a) Prove that $P(X_n \ge c \, n \quad infinitely \quad often)=1$ 
for all $c>0$. 

(b) Prove that for all $c>0$
\beann 
P(\limsup_{n \ra \infty} \frac{1}{n} \sum_{j=1}^n X_j \ge c) =1
\eeann 

(c) Prove that 
\beann 
P(\limsup_{n \ra \infty} \frac{1}{n} \sum_{j=1}^n X_j =\infty) =1
\eeann 

\end{enumerate}


\end{document}
\end

\newpage

(??) Let $l^\infty$ be the space of bounded 
real sequences $(c_n)_{n=0}^\infty$ with the usual sup norm. 
Let $C((-1,1))$ be the space of continuous functions on the {\bf open}
interval $(-1,1)$.
For $0<r<1$ we define a semi-norm by 
\beann
||f||_{\infty,r} = \sup_{|x| \le r} |f(x)|  
\eeann 
This family of semi-norms defines a topology on 
$C((-1,1))$ in the usual way.
Given $c=(c_n)_{n=0}^\infty \in l^\infty$ we define
\beann
f_c(x)=\sum_{n=0}^\infty c_n x^n
\eeann
Prove that the map $c \ra f_c$ is 
a continuous map from $l^\infty$ into $C((-1,1))$. 


(??) Let $X$ be a locally compact, Hausdorff space. Let $\mu_n, \mu$
be inner regular Borel measures on $X$. Suppose that for each
$f \in C_c(X)$,
$$\lim_{n \rightarrow \infty} \int f d \mu_n = \int f d \mu$$
 
\no (a) Prove that 
$$\liminf_{n \rightarrow \infty} \mu_n(U)  \ge \mu(U)$$
for all open sets $U$.

\no (b) Give an example to show that $>$ is possible in (a).

(??) Let $H$ be a separable Hilbert space over the reals, 
and let $\{e_n\}_{n=1}^\infty$ be an orthonormal basis for it.
Any element of $H$ can be written in a unique way as 
$\sum_{n=1}^\infty c_n e_n$ where $c_n$ is a square summable sequence of 
real numbers.
Define bounded linear operators $S_k$ by
\beann 
S_k(\sum_{n=1}^\infty c_n e_n) = \sum_{n=1}^\infty c_{n-k} e_n
\eeann
Intuitively, $S_k$ shifts the sequence $c_n$ to the left and 
chops off the first $k$  terms. 

(a) Compute the adjoint operator $S_k^*$.  See ...

(b) Show $S_k$ converges to zero in the strong operator topology.

(c) Show $S_k^*$ does not converge to zero in the strong operator topology.



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\centerline{\bf OLD Math 523b - Midterm}

\no {\bf Do four of the five problems. } 

\bigskip

\no 1. Let $<a_n>_{n=0}^\infty$ be a sequence of real numbers with the
following property. If $N$ is a nonnegative integer and 
$c_0,c_1, \cdots, c_N$ are real numbers such that 
$\sum_{n=0}^N c_n x^n \ge 0$ for all $x \in [0,1]$,
then $\sum_{n=0}^N c_n a_n \ge 0$.
Prove there exists a unique Borel measure $\mu$ on $[0,1]$ such that 
$$a_n = \int _{[0,1]} \, x^n d \mu $$

\no 2. Let $1 \le p \le \infty$. Let $f \in L^p(R)$, $g \in L^1(R)$.
Prove that for almost all $x$,
$$\int \, |f(x-y) g(y) | dy < \infty $$
and the function 
$$h(x) = \int \, f(x-y) g(y) dy < \infty $$
is in $L^p(R)$ and $||h||_p \le ||f||_p ||g||_1 $.
Hint : For $p<\infty$ we can show something is in $L^p$ by showing
it defines an element of the dual of $L^q$.

\no 3. Let $X$ be a compact Hausdorff space. Let $\mu$ be an inner regular 
Borel measure on $X$. Show there exists a unique compact set $K$ such that 
$\mu(\tilde K)=0$ and for any compact set $H$, $\mu(\tilde H)=0$ 
implies $K \subset H$. ($K$ is sometimes called the support of $\mu$.)
Hint : Consider the open sets $O$ such that $\mu(O)=0$. What can you
say about the union of all these open sets?

\no 4. Let $X$ be a locally compact, Hausdorff space. Let $\mu_n, \mu$
be inner regular Borel measures on $X$. Suppose that for each
$f \in C_c(X)$,
$$\lim_{n \rightarrow \infty} \int f d \mu_n = \int f d \mu$$
 
\no (a) Prove that 
$$\liminf_{n \rightarrow \infty} \mu_n(U)  \ge \mu(U)$$
for all open sets $U$.

\no (b) Give an example to show that $>$ is possible in (a).

\no 5. Let $X$ and $Y$ be Banach spaces. Let $F : X \times Y \rightarrow R$
be a function which is continuous in each argument separately and bilinear.
(These properties mean that if we fix an $x \in X$, then the function
$f_x(y) = F(x,y)$ is a linear continuous function from $Y$ to the reals,i.e.,
it is a bounded linear functional. A similar statement holds if we fix 
a $y \in Y$.)
Prove that $F$ is continuous on $X \times Y$. The topology on $X \times Y$ is  
the product topology. To show $F$ is continuous it suffices to show that
if $x_n \rightarrow x$ in $X$ and  $y_n \rightarrow y$ in $Y$, then
$F(x_n,y_n) \rightarrow F(x,y)$.














\bigskip
\bigskip

\centerline{\bf OLD Math 523b - Final}

\medskip

\no Do {\bf SIX} of the problems. Do not turn in more than six. 

\no 1. Let $X$ and $Y$ be Banach spaces. Let $T_n$ be a sequence
of bounded linear operators from $X$ to $Y$ such that 
$\sup_n ||T_n|| < \infty$. Suppose there is a dense subset $E$
of $X$ such that $\lim_{n \rightarrow \infty} T_n x$ exists for
all $x \in E$. Prove that $\lim_{n \rightarrow \infty} T_n x$ exists for
all $x \in X$.

\no 2. Let $\{e_n\}_{n=1}^\infty$ be an orthonormal set in a Hilbert
space. (It need not be an orthonormal basis.) Let
$$M= \{\sum_{n=1}^\infty \alpha_n e_n : 
  \sum_{n=1}^\infty |\alpha_n|^2 < \infty \}$$
Prove that $M$ is a closed subspace of the Hilbert space.

\no 3. Let $E$ be a subset of $R$ with Lebesgue measure zero.
Let $F= \{ (x,y):x+y \in E\}$. Prove that $F$ has Lebesgue measure zero.
(You may assume that it is measurable.)

\no 4. Let $\mu$ and $\mu^\prime$ be regular Borel measures on $[0,1]$
with $\mu([0,1])=\mu^\prime([0,1])=1$ and
$$\int_{[0,1]} \, e^{nx} d\mu =  \int_{[0,1]} \, e^{nx} d\mu^\prime$$
for $n=1,2,3,\cdots$. Prove that $\mu=\mu^\prime$.

\no 5. Let $\mu$ be a regular Borel measure on a locally compact 
Hausdorff space $X$. 
Let $E$ be a measurable set with finite measure. Let $\chi_E$ be 
the characteristic function of $E$. 
Prove that there is a sequence of continuous functions $f_n$ such
that $f_n \rightarrow \chi_E$ $\mu$ a.e. (You may NOT use the fact that
the continuous functions are dense in $L^1(X,\mu)$.)

\no 6. Let $\mu$ be a $\sigma$-finite measure on $X$ such that
$L^2(X,\mu) \subset L^1(X,\mu)$. Prove that $\mu$ is a finite
measure.

\no 7. Let $X$ be a compact Hausdorff space and $f_n \in C(X)$.
Show that $f_n$ converges weakly to zero if and only if 
$sup_n ||f_n|| < \infty$ and for every $x$, 
$\lim_{n \rightarrow \infty} f_n(x) = 0 $.

\no NOT USED 3. (hard) Let $\mu$ be a finite Borel measure on the 
closed disc $D$ in the plane. Prove that 
$$\int_D {d \mu(x) \over |x-y|} < \infty$$
for almost all $y \in D$ with respect to Lebesgue measure.
 
\no NOT USED 2. Let $\mu$ be a measure on $X$ and let $f,g$ be measurable
functions on $X$. Prove that there exits a unique Borel
measure $\nu$ on $R^2$ such that 
$$\nu([a,b] \times [c,d])= \mu(\{x:a \le f(x) \le b, c \le g(x) \le d\})$$

(Recall that the dual of $C(X)$ is the space of finite signed 
regular Borel measures on $X$.)
















































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\centerline {\bf Math 523a - Final - Fall 2012}

\bigskip

{\bf Do four of the following five problems.}

\bigskip

\begin{enumerate}

\item 
Let $x_n$ be an arbitrary sequence in $\reals$. Define 
\beann
f(x)=\sum_{n=1}^\infty \exp(-n^2 |x-x_n|) 
\eeann
Prove that $f(x)<\infty$ a.e. .  (The a.e. is with respect to Lebegue
measure).  

\item 
In this problem $X$ is $[0,\infty) \times [0,\infty)$ and $m$ is 
two dimensional Lebesgue measure.

\no (a) Let $f(x,y)=\exp(-(x+1)^2 y)$. Prove that $f$ is integrable 
on $X$. 

\no (b) Let $\lambda_n>0$ with $\lambda_n \ra 0$  as $n \ra \infty$.
Find 
\beann
\lim_{n \ra \infty} \int_X \exp(-(x+1)^2 (y+\lambda_n)) \, dm 
\eeann
Your answer should be a number and you should prove your result.

\item We proved in a homework problem that if $F$ is an increasing
function on $[a,b]$, then 
\beann 
\int_a^b F^\prime(x) \, dx \le F(b)-F(a)
\eeann
Give an example of an increasing function $F(x)$ on $[0,1]$ such that 
for all $0<a<b<1$ we have 
\beann 
\int_a^b F^\prime(x) \, dx < F(b)-F(a)
\eeann
You should justify your answer. 

\newpage

\item Let $g \ge 0$ be a Lebesgue measurable function on $\reals$. 
Let $m$ denote Lebesgue measure on $\reals$ and define another measure
by $\mu = g m$, i.e., 
\beann
\mu(E) = \int_E g \, dm
\eeann
Find the Lebesgue-Radon-Nikodym decomposition of $m$ with respect to 
$\mu$, i.e., express $m$ as $f \mu + \lambda$ where $f$ is a non-negative
measurable function (so $f \mu$ is absolutely 
continuous with respect to $\mu$) and $\lambda \perp \mu$. 
Cautions: $g$ can be zero on a set with non-zero Lebesgue measure. 
I am not asking for the decomposition of $\mu$ with respect to $m$.

\item We take our measurable space to be $\naturals$ with the 
$\sigma$-algebra of all subsets of $\naturals$. We define a measure
by
\beann
\mu(E)=\sum_{k \in E} 2^{-k} \quad for \quad E \subset \naturals
\eeann
Let $f_n$ and $f$ be functions on $\naturals$, all of which are 
integrable with respect to $\mu$. 
Consider the three modes of convergence:

(a) $f_n$ converges to $f$ point-wise 

(b) $f_n$ converges to $f$ in measure

(c) $f_n$ converges to $f$ in $L^1$ norm. 

There are six possible implications ((a) $\Rightarrow$ (b), 
(a) $\Rightarrow$ (c),etc).  
Prove the ones that are true and give counterexamples for those that are false. 

\end{enumerate}

\bigskip

\centerline {\bf Math 523b - Take home midterm}

\bigskip

\begin{enumerate} 

\item 
Let $\e$ be the set of even polynomials which we consider as 
functions on $[0,1]$. For $1 \le p \le \infty$
this is a subspace of $L^p([0,1],\borel,m)$
where $m$ is Lebesgue measure and $\borel$ are the Borel sets.
Determine (with proof) the closure of $\e$ in 
$L^p$ for $1 \le p \le \infty$. 

\item Let $(X,\M,\mu)$ be a measure space. We abbreviate 
$L^p(X,\M,\mu)$ as $L^p$. Let $1 <p,r < \infty$. 
Fix a function $f \in L^p$ and define a linear operator $T$ by 
$T g = f g $, where $g \in L^r$. Find a condition on $p,q,r$ that
implies $T g \in L^q$ for all $g \in L^r$ and prove that $T$ is 
a bounded linear operator from $L^r$ to $L^q$ when this condition holds.

\item Let $P$ be the set of power series $\sum_{n=0}^\infty c_n x^n$
such that 
$$
\sum_{n=0}^\infty (n+1) |c_n| \le 1
$$ 
It is easy to show that such a series converges 
uniformly on $[0,1]$ and so defines
a continuous function. So $P \subset C([0,1])$. 
(You don't have to show this). 
Prove that the closure of $P$ in $C([0,1])$ is compact. 
The norm is the usual sup norm.

\item 
Let $1<p<\infty$ and let $q$ be the conjugate exponent given 
by $1/p+1/q=1$. 
Let $b_n$ be a real valued sequence such that 
$\sum_{n=1}^\infty a_n  b_n$ converges for all real sequences $a_n$
in $l^p$. Prove that $b_n$ is in $l^q$. 
Hint: consider the linear functionals $f_N$ on $l^p$ defined by 
$$
f_N(\{a_n\}_{n=1}^\infty)= \sum_{n=1}^N b_n a_n
$$

\newpage

\item Consider $L^2=L^2([0,\infty),\borel,m)$ where $m$ is Lebesgue measure
and $\borel$ is the Borel sets.
Let 
$$
V_n = \{ f \in L^2 : f=0 \, a.e. \, on \, (n,\infty)\}
$$
($f=0 \, a.e. \, on \, (n,\infty)$ means that 
$m(\{x \in (n,\infty):f(x) \neq 0 \})=0$. )
It is not hard to show $V_n$ is closed. You may assume this fact without
proving it. Let $P_n$ be the orthogonal projection onto $V_n$. 
\begin{enumerate}
\item Find an explicit formula for $P_n$, i.e., given $ f \in L^2$
what is $P_n f$?
\item It is natural to expect that $P_n$ converges to 
the identity $I$ in some sense. 
Here are two possible senses in which it could converge.
\begin{enumerate}
\item $||P_n-I|| \ra 0$. (Here the norm is the operator norm.)
\item $P_n$ converges to $I$ in the strong operator topology, i.e., 
for all $f \in L^2$, $||P_n f - f|| \ra 0$. (Here the norm is the norm 
in $L^2$.)
\end{enumerate}
For each of these sense, either prove or disprove that $P_n$ converges to 
$I$ in this sense. 

\end{enumerate}

\end{enumerate}

\bigskip


\centerline {\bf Math 523b - Homework 1} 

\bigskip

\begin{enumerate}

\item Let $X$ be a topological space and $A$ and $B$ subsets of it. 
For each of the following statements either prove it or disprove it 
by giving a counterexample. 
  \begin{enumerate}
  \item $\overline {A \cup B}= \overline{A} \cup \overline{B}$
  \item $\overline {A \cap B}= \overline{A} \cap \overline{B}$
  \end{enumerate}

\item Folland, problem 5, page 118.
Every separable metric space is second countable.

\item Folland, problem 10, page 118.
connectedness

\item Folland, problem 15, page 123.
Continuous function on closed set which vanishes on boundary has 
continuous extension.

\item Folland, problem 16, page 123.
Hausdorff space: 2 cont functions agree on dense set implies they are 
equal.

\item Folland, problem 22, page 123.
completeness of functions between two complete metric spaces.

\item Folland, problem 28, page 124.
Quotient topology.

\item Folland, problem 40, page 130.
In countably compact space every sequence has a cluster point.

\item Folland, problem 43, page 130.
compact but not sequentially compact

\end{enumerate}


\bigskip

\centerline {\bf Math 523b - Homework 2} 

\bigskip

\begin{enumerate}

\item Problem 56, p. 135 in Folland. 
metric for uniform convergence on compact sets

\item Problem 60, p. 138 in Folland. 
countable product of sequentially compact spaces is sequentially compact

\item Problem 63, p. 138 in Folland. 
AA: range of integral operator is pre-compact

\item Problem 67, p. 142 in Folland. 
one point compactifications

\item Problem 68, p. 142 in Folland. 
SW applied to C(X x Y)

\item Let $X$ be a locally compact Hausdorff space. Let $\mu$
be a Borel measure on $X$. We say it is a Radon measure if it satisfies
these three properties:
\begin{enumerate}
\item (local finiteness) For all compact sets $K$, $\mu(K) < \infty$. 
\item (outer regularity) For all Borel sets $E$
$$
\mu(E)=\inf \{ \mu(U) : U \, open, E \subset U \}
$$
\item (inner regularity) For all Borel sets $E$
$$
\mu(E)=\sup \{ \mu(K) : K \, compact , K \subset E \}
$$
\end{enumerate}
Prove that if $\mu$ is such a measure, then $C_c(X)$, the continuous
functions with compact support, are dense in $L^1(X,\mu)$.

\item Let $X$ be a separable compact metric space. Prove that 
$C(X)$ (real or complex valued) is separable. Hint: Let 
$x_k$ be a dense sequence in $X$ and consider the 
algebra of functions generated by the functions
$$
f_{k,n} = \max \{ 0, 1-n \rho(x,x_k)\}
$$
where $\rho$ is the metric. 

\end{enumerate}

\bigskip

\centerline {\bf Math 523b - Homework 3} 

\bigskip

\begin{enumerate}


\item Problem 2, p. 154 in Folland. 
equivalent defs of operator norm

\item Problem 9, p. 155 in Folland. 
$C^k[0,1]$.

\item Problem 21, p. 160 in Folland. 
Dual of product space.

\item Problem 22, p. 160 in Folland. 
Adjoint

\item Problem 25, p. 160 in Folland. 
$X^*$ separable implies $X$ separable

\item Problem 34, p. 164 in Folland. 
closed graph theorem 

\item Problem 37, p. 165 in Folland. 
UBP

\item Problem 39, p. 165 in Folland. 
bilinear map and UBP

\item This problem is a special case of a general theorem we will
prove later. Let $c_0$ denote the Banach space of sequences 
which converge to $0$ with the norm
$$
|| (x_n)_{n=1}^\infty||_\infty = \sup_n |x_n|
$$
Let $l^1$ be the Banach space of absolutely summable sequences
with the norm
$$
|| (x_n)_{n=1}^\infty||_1 = \sum_{n=1}^\infty |x_n|
$$
Our goal is to show that the dual of $c_0$ can be identified with 
$l^1$ in a natural way. 

\no (a) Given $x=(x_n)_{n=1}^\infty \in l^1$, define a linear functional
$\phi_x$ on $c_0$ by 
$$
\phi_x((y_n)_{n=1}^\infty) = \sum_{n=1}^\infty x_n y_n
$$
Prove that $\phi_x$ is a bounded linear functional whose norm equals
$||x||_1$. Thus the map $x \ra \phi_x$ is an isometry of 
$l^1$ into the dual of $c_0$. \

\no (b) Prove that this isometry is onto. 


\end{enumerate}

\bigskip

\centerline {\bf Math 523b - Homework 4} 

\bigskip

\begin{enumerate}


\item 
\begin{enumerate}
\item Let $X,Y$ be Banach spaces. Let $\{T_n\}_{n=1}^\infty$ and $T$ 
be bounded linear operators from $X$ to $Y$. We say that $T_n$ 
converges to $T$ in the strong operator topology 
if $T_n x \ra T x$ in norm in $Y$ for all $x \in X$.
Prove that when $T_n$ converges to $T$ in the strong operator 
topology we have
$$ 
\sup_n ||T_n|| < \infty
$$
\item 
Prove that every weakly convergent sequence in $X$ is bounded. 
\end{enumerate}

\item 

\begin{enumerate}
\item Prove that a Hilbert space is separable if and only if it has 
a countable basis. {\bf You don't have to turn in this part.}
\item Let $H$ be a separable infinite dimensional Hilbert space over
$\complex$. Let $\{e_n \}_{n=1}^\infty$ be an o.n. basis. 
Let $t_{n,m} \in \complex$ for $n,m \in \naturals$. 
Suppose that 
$$
\sum_{n,m} |t_{m,n}|^2 < \infty
$$
Prove there is a bounded linear operator $T$ on $H$ such that 
$t_{m,n} = (T e_n,e_m)$. You can think of $t_{m,n}$ as the infinite 
dimensional matrix of $T$. 
(This can be proved under weaker assumptions on the $t_{n,m}$.)
\end{enumerate}


\item Problem 61, p. 178 in Folland.
o.n. basis for $L^2(X \times Y)$.

\item Problem 63, p. 178 in Folland. 
Unit sphere weakly dense in unit ball in a Hilbert space.

\item This is a generalization of Holder's inequality. 
Let $p_1,p_2,\cdots,p_n > 0$. Let $f_i \in L^{p_i}$ for $i=1,2,\cdots,n$. 
Prove that 
$$
|| f_1 f_2 \cdots f_n ||_1 \le \prod_{i=1}^n ||f_i||_{p_i}
$$
under a suitable condition on the $p_i$. You should of course state
what the condition is.  

\item Problem 5, p. 186 in Folland. 
$L^p$, $L^q$ inclusions

\bigskip

\centerline{\bf \large The following are not to be turned in }

\item Problem 57, p. 177 in Folland. 
Adjoint in Hilbert space

\item Problem 7, p. 186 in Folland. 
$||f||_\infty$ as limit of $||f||_p$. 

\end{enumerate}

\bigskip


\centerline {\bf Math 523b - Homework 5} 

\bigskip

\begin{enumerate}

\item Folland problem 19, p. 192
Dual of $l^\infty$ is not $l^1$.


\item Folland problem 20a, p. 192. Note that only part (a) is assigned.
weak convergence in $L^p$

\item Folland problem 32, p. 197
bound on norm of integral operator

\item Folland problem 36, p. 199
weak $L^p$

\item Folland problem 39, p. 199
distribution function

\item Folland problem 2, p. 215
support of a measure

\item Folland problem 4, p. 215
Baire sets

\item 
This problem gives an example to show that the measure
associated with a postive linear functional on $C_c(X)$ by
the Riesz representation theorem need not be regular. 
Let $X$ be the plane with the following topology. A set is open
if and only if its intersection with every vertical line is an 
open subset of that line. Show that $X$ is a locally compact Hausdorff
space. Let $f \in C_c(X)$. Show that there are only finitely
many $x$ such that $f(x,y) \ne 0$ for at least one $y$. 
Let $x_1,x_2,\cdots x_n$ be these $x$'s. Define 
$$F(f) = \sum_{j=1}^n \, \int_{- \infty}^\infty \, f(x_j,y) dy$$
Obviously $F$ is a positive linear functional, and so is represented
by a measure $\mu$. Let $E$ be the $x$-axis. Show that 
$\mu(E)= \infty$ but $\mu(K)=0$ for every compact $K \subset E$.




\end{enumerate}

\bigskip

\centerline {\bf Math 523b - Homework 6 (Last one!) } 

\bigskip

\begin{enumerate}

\item Folland problem 22, p. 225
weak convergence in $C_0$ equivalent to pointwise convergence and 
bounded

\item Folland problem 24, p. 225
Examples involving vague convergence

\item Folland problem 27, p. 225
Dual of $C^k[0,1]$.

\item  In a homework problem you showed that the dual of $c_0$, 
the space of sequences that converge to zero with the sup norm, 
is isometric to $l^1$. Now let $c$ denoted the space of real-valued 
sequences that converge to some finite real number. 
Given $y=\{y_n\}_{n=1}^\infty \in l^1$ we can still define a linear
functional $f_y$ by 
$$
f_y(\{x_n\}_{n=1}^\infty)=\sum_{n=1}^\infty x_n y_n
$$
and the same argument as for $c_0$ proves this is a bounded linear
functional with norm equal to the $l^1$ norm of $y$. 
So we have an isometry from $l^1$ {\bf into} $c^*$.
\begin{enumerate} 
\item The Riesz representation theorem says that the dual of $c_0$ is 
isometric with a space of measures. Explain briefly how this is 
related to the assertion that $c_0^*$ is isometric with $l^1$. 
\item Prove that the isometry above from $l^1$ into $c^*$ is not onto.
\item Prove that there is a linear function $f_\infty \in c^*$ so 
that every linear functional in $c^*$ can be written as 
$f_y + a f_\infty$ for some $y \in l^1$ and some real number $a$. 
(So the range of the isometry has co-dimension 1.)
\end{enumerate} 


\item Let $f(x)$ be a Borel measurable real-valued function on $[0,1]$ 
with \\
$\int_0^1 f^2(x) \, dx < \infty$. Let $\{U_n\}_{n=1}^\infty$
be an independent sequence of random variables, each of which is uniformly 
distributed on $[0,1]$, i.e., the distribution $P_{U_N}$ of $U_n$ is Lebesgue 
measure on $[0,1]$. Let 
\bea
S_n= {1 \over n} \sum_{i=1}^n f(U_i)
\nonumber
\eea
Prove that $S_n$ converges to the constant $\int_0^1 f(x) \, dx$
in probability.


\item 
Let $E_n$ be a sequence of events. Recall that 
\bea
\limsup E_n = \cap_{n=1}^\infty \cup_{k=n}^\infty E_k
\eea
In probability this event is usually denoted $E_n \, i.o.$
where i.o. stands for ``infinitely often.''
Let $X_n$ be an independent, identically distributed 
sequence of non-negative random variables. 
Prove that $EX_1 <\infty$ if and only 
if $P(X_n \ge n \, \, i.o.)=0$. 

\item Suppose $X_n$ are independent, identically distributed, and have 
finite second moment, i.e.,
$EX_n^2 < \infty$. We proved a weak law of large numbers 
which says that for any
$\epsilon > 0$, the probability 
$$P(|{1 \over n} \sum_{k=1}^n X_k - \mu| > \epsilon)$$
converges to zero as $n \rightarrow \infty$, but we didn't say anything 
about how fast it converges.

\no (a) Use the proof from class of the weak law for the
case of finite second moment to show it converges to zero at least 
as fast as $1/n^p$ for some power $p$. (I.e., show that the probability 
is $\le c/n^p$ for some constant $c$, which can depend on $\epsilon$.)
You should find the biggest $p$ you can. 

\no (b) Now suppose that you also know that $EX_n^4 < \infty$.
Prove that $P(|{1 \over n} \sum_{k=1}^n X_k - \mu| > \epsilon)$
converges to zero faster than your bound from (a), i.e., with a bigger $p$. 


\end{enumerate}
\end{document}
\end


















\item 
Let 
\beann
c_0=\{ (x_k)_{k=1}^\infty : \lim_{k \ra \infty} x_k =0\} \\
l_1=\{ (x_k)_{k=1}^\infty : \sum_{k=1}^\infty |x_k| <\infty \} 
\eeann
with their usual norms. Recall that there is a natural isometry 
between $l^1$ and the dual of $c_0$.
Let $x^n=(x^n_k)_{k=1}^\infty$ be a sequence of elements of $l^1$
such that 

(1) $\sup_n ||x^n||_1 < \infty$

(2) For all $k$, $\lim_{n \ra \infty} x^n_k =0$ 

Prove that $x^n$ converges to the zero sequence in the weak * 
topology.

