**Real Analysis**

**Instructor:**Guanghua Ji

**TA:**Lingyu Li, Ting Chen

**Grading:**Be based on homework(30%) and final exam(70%).

**Schedule: **THU 1-2; FRI 5-6.

**References**:

1, 周民强, 实变函数论,北京大学出版社, 2012.

2, 郭懋正, 实变函数与泛函分析, 北京大学出版社, 2009.

3, 胡适耕, 实变函数, 高等教育出版社, 2014.

4, E. M. Stein, R. Shakarchi, Real Analysis, Princeton University Press, 2006

5, R. L. Wheeden, A. Zygmund, Measure and integral, an introduction to real analysis, CRC Press, 2015.

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**Chapter 1. Preliminaries**

2/22,§1.1, Sets and Mappings:I, Sets. II, Sequences of sets. III, Mappings. IV, Countable sets. HW

2/24, §1.2, Topology of R^n:I, Points sets. II, Open, closed compact sets.

3/2, III, Construction of open sets. IV, Continuity.

3/3, V, Distance functions. VI, Important theorems: Cauchy converegence thm, Heine-Borel thm(finite cover thm), Cantor's intersection thm(A decreasing nested sequence of non-empty compact subsets has non-empty intersection), Bolzano-Weierstrass thm, Continuos functions on compect sets.

**Chapter 2. Measure Theory**

3/9,§2.1, The exterior measure:I, Defintion. II, Properties. HW: p.37. 1,5,6.

3/10,§2.2, Lebesgue Measurable sets:I, Defintion and properties: open set, null set, countable union, closed set, complement, countable intersection. II,Cantor Set. III, A nonmeasurable set. HW: p.39. 7,8.

3/16,§2.3, Lebesgue Measures:I, Countable additivity. II, Sequence of measurable sets. III, Equivalent definitions of measurability. IV, Continuous transformation. V, Product measure. HW: 11,13,15,16.

3/17,§2.4,Abstract Measure Theory: I, Sigma Algebra and Borel Sets. II, Measure space and measure.

3/23, III, Exterior measures and caratheodory's thm. IV, The Construction of exterior measure. HW

**Chapter 3. Measurable Functions**

3/24,§3.1, Elementary Properties of Measurable Functions:I, Definition. II, Approximation by simple functions or step functions. HW: p42, 17,18,25,26.

3/30,§3.2, Mode of convergence(Diagrams):I, a.e.c and n.u.c: Egorov thm, converse thm of Egorov thm. II, a.e.c, c.m and Sub. a.e.c.: Lebesgue thm, Riesz's thm. HW

3/31, III, Cauchy sequence in measure.§3.3, Lusin's Theorem:measurable functions and continuous func.

**Chapter 4. Integration Theory**

4/6,§4.1, Definition of the Integral:I, Simple functions II, nonnegative measurable funtions: monotone convergence theorem, Fatou lamma. HW: p90, 4,5.

4/7, III, general measurable functions: linearrity, aditivity, monotonicity, traiangle inequality, absolutly continuity. HW: p.91, 9,10.

4/13,§4.2,Convergence Theorem:I, Monotone convergence theorem and Fatou lemma for general fucntions. II, Dominated convergence thm, bounded conv. thm. III, Vitali convergence thm. HW: 11,12,16.

4/14,§4.3,Riemann and Lebesgue integrals:I, Riemann integrable implies Lebesgue integrable. II, Sufficient and necessary conditions of Riemann integral. HW

4/20,§4.4, Fubini's theorem:I, Fubini's thm. II, Tonelli's thm. III, Geometric meaning of Integration. HW: p.93, 19,21.

**Chapter 5.Differentiation Theory**

5/19,§6.1, Differentiation of IntegralI, The Hardy-Littlewood maximal function: averaging problem, Vitali covering lemma. II, The Lebesgue differentiation theorem: locally integrable, Lebesgue set. HW: p146, 4,5,9.

5/25,§6.2, Monotone functions:I, Continuity of Monotone Functions. II, The Vitali Covering Lemma. III, Differentiability of Monotone Functions. HW

5/26,§6.3, Functions of bounded variationI, Defiiniton II, Jordan decomposition III, Lebesgue Theorem. HW: p147, 11,16.

6/1, §6.4, Absolutely continuous functionsI, Definition II, Newton-Leibniz-Lebesgue Theorem: sufficient and necessary conditions. HW: p150, 23,32.

6/2, §6.5, Riemann-Stieltjes IntegralI, Definitons II, Integration by parts III, Applications in number theory.

**Chapter 6.L^p Spaces**

4/27,§5.1, L^P: I, Definition, exxential bounded. II, Basic inequality: Young's inequality, Holder's inequality, Schwartz's inequality, Minkowshi's inequality. HW: p193, 1,2.

4/28,§5.2, Complete and Separable;III, Complete: metric space, normed linear space. IV, Approximation: Dense subspaces(simple, continuous fun. of compact support, step fun. of compact support). V, Separable metric spaces. HW

5/4,§5.3, L^2I, Hilbert space: inner space. II, Orthogonal system: countable. HW: p.194, 3,5.

5/5,§5.4, Orthonormal Basis:III, Fourier series: Bessel's inequality, Riesz-Fischer thm.IV, Orthonormal basis: five equivalent characterizations: complete, maximal, dense, Parseval's formula, inner product formula. HW: 9.

**Chapter 7.Fourier Analysis**

5/11,§7.1, ConvolutionI, Convolution functions. II, Approximations of the identity. HW: p94, 23.p253, 1.

5/12,§7.2, Fourier transform on L^1:I,Fourier transformon L^1: elementary properties(linear, BC, Riemann-Lebesgue, Translation, Dilation, Rotation, Shifting hats, Convolution), Fourier inversion formula.

5/18, §7.3, Fourier transform on L^2:II, Schwartz functions, Differentiation properties. III, Fourier transform on L^2: Plancherel's formula.

6/8,§7.4, Fourier coefficients and seriesI,VI, Periodic functions L^2(T^n) II, Poisson sumation formula.

6/9,§7.5, Pointwise convergence of Fourier series.I, Dirichlet kernel II, Dirichlet-Jordan Thm.

6/15, Recitation

6/16, Recitation

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**Homework Supplements**

**Yau College Student Maths Contests**

**Final Exam of Real Analysis**