分类： Qualify Eaxm

Analysis Qualifying Examination(UCLA 2009)

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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)<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.

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.

 

 

 

 

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