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分类： Measure theory

  • UCLA 2009 Analysis Qualifying Exam：几个分析题的共同结构

    旧博客原文

    原题：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)<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.

     

     

     

     

    补充说明

    以下是新整理的中文说明；上方旧博客原文保持不变。

    这组题表面上分散：有测度论、Hilbert 空间弱拓扑、Banach 空间可分性、函数空间闭性。但它们的共同核心是：把抽象对象转化为可数逼近、积分层分解或闭性问题。

    UCLA 2009 Analysis Qualifying Exam：几个分析题的共同结构
    测度层分解、弱拓扑逼近、对偶可分性和闭子空间完备性，是这组分析资格考试题背后的共同工具。

    1. Layer-cake 与 Fubini

    对非负函数 $f$，常用恒等式是

    $$f(x)=\int_0^\infty \mathbf 1_{\{f(x)>t\}}\,dt.$$

    积分后用 Fubini 交换次序，就得到 layer-cake representation。很多看似是范数不等式的题，其实都可以化成集合层面的包含关系。

    2. Hilbert 空间单位球的弱拓扑

    在无限维 Hilbert 空间中，单位球的弱闭包现象和强拓扑非常不同。给定 $\|x\|\le1$，可以在与有限多个测试向量正交的方向上添加一个小向量，使范数变成 $1$，同时不改变这些测试函数的值。于是单位球中的点可被单位球面弱逼近。

    3. 可分性与对偶

    若 $X^*$ 可分，则 $X$ 可分。证明思路是从 $X^*$ 中取可数稠密集 $\{f_n\}$，再为每个有限线性条件选择近似达到范数的点。Hahn-Banach 的思想隐藏在这里：对偶空间的可数信息足以区分 $X$ 中的点。

    4. 闭子空间的完备性

    若 $L^2(D)$ 中的全纯函数子空间在 $L^2$ 收敛下保持全纯，那么它就是闭子空间，从而完备。关键是用局部估计把 $L^2$ 收敛提升为紧子集上的一致收敛，再由 Weierstrass 定理保全全纯性。