# Analysis Qualifying Examination(UCLA 2009)

1.

Let $f,g$ be real-valued integrable functions on a measure space $(X,B,\mu)$,and define:

$F_t=\{x\in X:f(x>t)\},G_t=\{x\in X:g(x)>t\}$.

Prove:

$\int|f-g|d\mu=\int_{-\infty}^{\infty}\mu((F_t-G_t)\cup(G_T-F_t))dt$.

proof:

by Fubini theorem(cake representation theorem in fact):

$\int|f-g|=\int_{0}^{\infty}\mu(\{x||f-g|(x)>t\})dt\\ \displaystyle =\int_{-\infty}^{\infty}\mu(\{x|f(x)>t>g(x)\})+\mu(\{x|f(x).

(there is a geometric heuristic,strict proof is due to fubini theorem)

Q.E.D.

2.

Let $H$ be a infinite dimensional real Hilbert space.

a)Prove the unit sphere $\{x\in H:||x||=1\}$ of $H$ is weakly dense in the unit ball $B=\{x\in H :||x||\leq 1\}$ of $H$.

b)Prove there is a sequence $T_n$ of bounded linear operator from $H$ to $H$ such that $||T_n||=1$ for all n but $lim T_n(x)=0$ for all $x\in H$.

proof:

by Zorn lemma there is a orthogonal bases $\{e_i\}$.

to proof a),suffice to proof:$\forall x,\exists x_n,\forall y\in H,\lim_{n \to \infty}=$.

this can be done by look at the expansion $y=\sum_{i}e_i$.due to the Cauchy inequality,there is a freedom of choice the coefficient $$ for $i>>n$.the choice will lead a).

b) is trivial due to a).

Q.E.D.

3.Let $X$ be a Banach space and let $X^*$ be it dual Banach space.Prove that if $X^*$ is separable then $X$ is separable.

proof:

we know $X^*$ is the space consist with bounded(continued) linear functional on $X$.

for $f\in X^*$,$||f||_{X^*}=\sup_{x\in B}||x||$,so due to $X^*$ is separable.there is a countable dense set $I$ in $X^*$.i.e. $\forall f\in X^*, \forall \epsilon >0,\exists f_{\epsilon}\in I,||f-f_{\epsilon}||_{x^*}<\epsilon$.we equip a member of $X$ to $f_{\epsilon}$ by $H:I \to X,H(f)=x,x=sup_{x\in B}||f(x)||$,$\hat I=Im(I)$.

On the other hand,$\forall x\in X$ we construct a functional $l_x$.$l_x(y)=||y|| iff y=cx,c\in R,or ,l_x(y)=0$.so it is obviously to show $\hat I$ is dense in $X$.

Q.E.D.

5.Let $I=I_{0,0}=[0,1]$ be the unit interval,and for $n=0,1,2,...$ and $0\leq j \leq 2^n-1$,let:

$I_{n,j}=[j2^{-n},(j+1)2^{-n}]$.

For $f\in L^1(I,dx)$ define $E_nf(x)=\sum_{j=0}^{2^n-1}(2^n\int_{I_{n,j}}fdt)\chi_{I{n,j}}$.

Prove that if $f\in L^1(I,dx)$ then $lim_{n\to \infty}E_nf(x)=f(x)$ a.e. in I.

proof:

…gap…

10.Let $D$ be the open unit disc and $\mu$ be Lebesgue measure on $D$.let $H$ be the subspace of $L^2(D,\mu)$ consisting of holomorphic functions.Show that $H$ is complete.

proof:maximum norm principle.to show $u_n$ is closed uniformly coverage.so is harmonic and $L^2$ is due to $L^2$ itself is complete.

Q.E.D.