SYLLABUS FOR THE REAL ANALYSIS QUALIFYING EXAM
PhD实分析资格考试提纲
Stanford University Mathematics Department
斯坦福数学系
Set Theory. Countable and uncountable sets , the axiom of choice , Zorn’s lemma.
集合论. 可数和不可数集, 选择公理,佐恩(Zorn)引理.
Metric spaces. Completeness ; separability ; compactness ; Baire category ; uniformcontinuity ; connectedness; continuous mappings of compact spaces.
度量空间. 完备性;可分性;稠密;贝利(Baire)范畴;一致连续性;连通性;紧集的连续变换
Functions on topological spaces. Equicontinuity and Ascoli’s theorem; the Stone-Weierstrass theorem; topologies on function spaces; compactness in function spaces.
拓扑空间的函数. 同等连续性和Ascoli’s定理; Stone-Weierstrass定理;函数空间拓扑;函数空间的紧性
Measure. Measures and outer measures; measurability and σ-algebras; Borel sets;extension of measures; Lebesgue and Lebesgue-Stieltjes measures; signed measures;absolute continuity and singularity; product measures.
测度. 测度和外测度;可测和σ-代数;博雷尔(Borel)集类;测度的扩张; 勒贝格(Lebesgue)测度和勒贝格-斯蒂杰(Lebesgue-Stieltjes)测度;符号(广义)测度;绝对连续性和奇点;乘积测度
Measurable functions. Properties of measurable functions; approximation by simple functions and by continuous functions; convergence in measure; Egoroff’s theorem; Lusin’s theorem; Jensen’s inequality.
可测函数. 可测函数的性质;简单和连续函数逼近;测度收敛;叶果洛夫(Егоров)定理;鲁津(Лузин)定理;詹森(Jensen)不等式
Integration. Construction and properties of the integral; convergence theorems; Radon-Nykodym theorem; Fubini’s theorem; mean convergence.
积分. 积分的构造和性质;各类收敛定理;拉东(Radon-Nykodym)定理;富比尼(Fubini)定理;平均收敛
Special properties of functions on the real line. Monotone functions; functions of bounded variation and Borel measures; absolute continuity; differentiation and integration; convex functions; semicontinuity; Borel sets; properties of the Cantor set.
一维实函数的特性. 单调函数;囿变函数;绝对连续性;微分和积分;凸函数;半连续性;博雷尔(Borel)集类; 康托(Cantor)集的性质
Elementary properties of Banach and Hilbert spaces. Lp spaces; C(X); completeness and the Riesz-Fischer theorem; orthonormal bases; linear functionals; Riesz representation theorem; linear transformations and dual spaces; interpolation of linear operators; Hahn-Banach theorem; open mapping theorem; uniform boundedness (or Banach-Steinhaus) theorem; closed graph theorem; Alaoglu theorem.
巴拿赫和希尔伯特(Banach and Hilbert)空间的基本性质. Lp空间; X上连续函数类C(X);完备性和黎兹-费舍尔(Riesz-Fischer)定理;标准直(正)交系; 线性泛函;黎兹(Riesz)表示定理; 线性转换和对偶空间;线性算子插值;哈恩-巴拿赫(Hahn-Banach)定理;开映射定理;一致有界(或者叫Banach-Steinhaus)定理;闭图定理; Alaoglu定理.
Basic harmonic analysis. Basic properties of Fouries series and the Fourier transform; Poission summation formula; convolution; mollification.
基本调和分析. 傅立叶(Fouries)级数和傅立叶变换的基本性质;泊松(Poission)叠加方程;卷积;缓和(?)
References
参考资料
The above topics are covered in the following:
上面这些内容请参考下面这些资料:
(1) Royden, Real Analysis, except chapters 8, 13, 15.
(1) Royden, Real Analysis, 不包括第8, 13, 15章.
(2) Dym and McKean, Fourier Series and Integrals, chapters 1 and 2, or Katznelson, Introduction to harmonic analysis, chapters 1, 2, and 4.
(2) Dym and McKean, Fourier Series and Integrals,第1,2章, 或Katznelson, Introduction to harmonic analysis, 第1,2,4章.
(3) Gilbarg and Trudinger, Elliptic Partial Di_erential Equations of Second Order, chapter 7.2 and 7.3.
(3) Gilbarg and Trudinger, Elliptic Partial Di_erential Equations of Second Order, 章节 7.2,7.3.