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):

$\int|f-g|=\int_{0}^{\infty}\mu(\{x||f-g|(x)>t\})dt\\ =\int_{-\infty}^{\infty}\mu(\{x|f(x)>t>g(x)\})+\mu(\{x|f(x)<t<g(x)\})dt\\ =\int_{-\infty}^{\infty}\mu((F_t-G_t)\cup(G_T-F_t))dt$.

(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:$\forall x,\exists x_n,\forall y\in H,\lim_{n \to \infty}<x_n,y>=<x,y>$.

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

b) is trivial due to a).

Q.E.D.

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