FOUNDATIONS OF MODERN APPLIED MATHEMATICS(近代应用数学基础)

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