# 实分析

SYLLABUS FOR THE REAL ANALYSIS QUALIFYING EXAM
PhD实分析资格考试提纲

Stanford University Mathematics Department

Set Theory. Countable and uncountable sets , the axiom of choice , Zorn’s lemma.

Metric spaces. Completeness ; separability ; compactness ; Baire category ; uniformcontinuity ; connectedness; continuous mappings of compact spaces.

Functions on topological spaces. Equicontinuity and Ascoli’s theorem; the Stone-Weierstrass theorem; topologies on function spaces; compactness in function spaces.

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.

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.

Integration. Construction and properties of the integral; convergence theorems; Radon-Nykodym theorem; Fubini’s theorem; mean convergence.

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.

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.

Basic harmonic analysis. Basic properties of Fouries series and the Fourier transform; Poission summation formula; convolution; mollification.

References

The above topics are covered in the following:

(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.