Table of Contents

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