近代应用数学基础

　　本书系统地介绍集合论、近世代数、点集拓扑、泛函分析、Fourier分析、分布理论、微分几何等近代应用数学的基本内容，及其在自然科学领域中的应用。书中强调对近代数学基本概念的理解、对重要论证方法的思路分析，以培养读者掌握并应用近代应用数学工具解决本专业的实际问题。20世纪初期至今的百余年中，数学科学与自然科学诸领域相辅相成，互相促进，彼此渗透，共同发展，使得数学科学成为当今各个科学领域中不可或缺的重要工具。因此介绍近代应用数学基本内容的教材已成当务之急，本书就起了这样的重要作用。

　　苏维宜，南京大学数学系教授，博士生导师。科研主攻方向是数学科学的重要分支——调和分析与分形分析。发表学术论文百余篇（其中半数以上发表在国内外SCI期刊上）。科研专著3部。完成国家重大基础研究项目（非线性科学）的子项目（分形分析）一项、国家自然科学基金面上项目十余项。培养获数学博士学位的研究生15名、获硕士学位的22名。指导博士后7名。科研成果卓著，是国内公认的本领域的学术带头人。在国际上有较大影响，多次主办国内外数学学术会议，并应邀作学术报告。教学方面，数十年中开设数学系基础课程、专业课程十余门，主持南京大学、江苏省、国家教学改革项目4项，主持国家精品课程《高等数学》十余年。编写本科生、研究生教材4本。在教育科研战线上辛勤耕耘52年，爱岗敬业，严谨治学。教学精益求精，深受学生爱戴，2015年荣获南京大学教学终身成就奖。

Preface
Chapter 1Set,Structure of Operation on Set
1.1Sets,the Relations and Operations between Sets
1.1.1Relations between sets
1.1.2Operations between sets 
1.1.3Mappings between sets
1.2Structures of Operations on Sets
1.2.1Groups,rings,fields,and linear spaces 
1.2.2Group theory,some important groups
1.2.3Subgroups,product groups,quotient groups
Exercise 1
Chapter 2Linear Spaces and Linear Transformations
2.1Linear Spaces 
2.1.1Examples
2.1.2Bases of linear spaces
2.1.3Subspaces and product/directsum/quitient spaces
2.1.4Inner product spaces
2.1.5Dual spaces
2.1.6Structures of linear spaces
2.2Linear Transformations
2.2.1Linear operator spaces
2.2.2Conjugate operators of linear operators
2.2.3Multilinear algebra
Exercise 2
Chapter 3Basic Knowledge of Point Set Topology
3.1Metric Spaces,Normed Linear Spaces
3.1.1Metric spaces
3.1.2Normed linear spaces
3.2Topological Spaces
3.2.1Some definitions in topological spaces
3.2.2Classification of topological spaces
3.3Continuous Mappings on Topological Spaces
3.3.1Mappings between topological spaces,continuity of mappings
3.3.2Subspaces,product spaces,quotient spaces
3.4Important Properties of Topological Spaces
3.4.1Separation axioms of topological spaces
3.4.2Connectivity of topological spaces
3.4.3Compactness of topological spaces
3.4.4Topological linear spaces
Exercise 3
Chapter 4Foundation of Functional Analysis
4.1Metric Spaces
4.1.1Completion of metric spaces
4.1.2Compactness in metric spaces
4.1.3Bases of Banach spaces
4.1.4Orthgoonal systems in Hilbert spaces
4.2Operator Theory
4.2.1Linear operators on Banach spaces
4.2.2Spectrum theory of bounded linear operators
4.3Linear Functional Theory
4.3.1Bounded linear functionals on normed linear spaces
4.3.2Bounded linear functionals on Hilbert spaces
Exercise 4
Chapter 5Distribution Theory
5.1Schwartz Space,Schwartz Distribution Space
5.1.1Schwartz space
5.1.2Schwartz distribution space
5.1.3Spaces ERn,DRn and their distribution spaces
5.2Fourier Transform on LpRn,1≤p≤2 
5.2.1Fourier transformations on L1Rn
5.2.2Fourier transformations on L2Rn
5.2.3Fourier transformations on LpRn,1<><>
5.3Fourier Transform on Schwartz Distribution Space
5.3.1Fourier transformations of Schwartz functions
5.3.2Fourier transformations of Schwartz distributions
5.3.3Schwartz distributions with compact supports
5.3.4Fourier transformations of convolutions of Schwartz distributions
5.4Wavelet Analysis
5.4.1Introduction
5.4.2Continuous wavelet transformations
5.4.3Discrete wavelet transformations
5.4.4Applications of wavelet transformations
Exercise 5
Chapter 6Calculus on Manifolds
6.1Basic Concepts
6.1.1Structures of differential manifolds
6.1.2Cotangent spaces,tangent spaces
6.1.3Submanifolds
6.2External Algebra
6.2.1(r,s)type tensors,(r,s)type tensor spaces
6.2.2Tensor algebra
6.2.3Grassmann algebra (exterior algebra)
6.3Exterior Differentiation of Exterior Differential Forms
6.3.1Tensor bundles and vector bundles
6.3.2Exterior differentiations of exterior differential form
6.4Integration of Exterior Differential Forms
6.4.1Directions of smooth manifolds
6.4.2Integrations of exterior differential forms on directed manifold M
6.4.3Stokes formula
6.5Riemann Manifolds, Mathematics and Modern Physics
6.5.1Riemann manifolds
6.5.2Connections
6.5.3Lie group and movingframe method
6.5.4Mathematics and modern physics
Exercise 6
Chapter 7Complimentary Knowledge
7.1Variational Calculus
7.1.1Variation and variation problems
7.1.2Variation principle
7.1.3More general variation problems
7.2Some Important Theorems in Banach Spaces
7.2.1StoneWeierstrass theorems
7.2.2Implicit and inversemapping theorems
7.2.3Fixed point theorems
7.3Haar Integrals on Locally Compact Groups
Exercise 7
References
Index