近代應(yīng)用數(shù)學(xué)基礎(chǔ)

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