本教材是學(xué)習(xí)泛函分析課程的一本入門教材，是針對中國學(xué)生編寫的一本英文教材，在選材上吸收了國外的優(yōu)秀本科生教材的一些精華；在編寫上考慮了與中國學(xué)生所具備的基礎(chǔ)知識銜接性，在充分地反映泛函分析中的核心內(nèi)容的前提下，突出重點；在內(nèi)容的處理上，體現(xiàn)了由淺入深，循序漸進的原則，用大量的例題對度量空間、賦范線性空間、線性算子與線性泛函、內(nèi)積空間與各種算子及它們的譜分解的概念、關(guān)系、性質(zhì)進行了演繹、推導(dǎo)與論證，
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Contents
Preface i
Introduction iii
List of Symbols vii
Chapter 1 Metric Spaces 1
1.1 Preliminaries 1
1.2 Definitions and Examples 6
1.3 Convergence of Sequences in Metric Spaces 12
1.4 Sets in a Metric Space 17
1.5 Complete Metric Spaces 25
1.6 Continuous Mappings on Metric Spaces 33
1.7 Compact Metric Spaces 38
1.8 Banach Fixed Point Theorem 46
Chapter 2 Normed Linear Spaces. Banach Spaces 57
2.1 Review of Linear Spaces 57
2.2 Norms in Linear Spaces 59
2.3 Examples of Normed Linear Spaces 65
2.4 Finite-Dimensional Normed Linear Spaces 77
2.5 Linear Subspaces of Normed Linear Spaces 83
2.6 Quotient Spaces 90
2.7 Weierstrass Approximation Theorem 94
Chapter 3 Inner Product Spaces. Hilbert Spaces 101
3.1 Inner Products 101
3.2 Orthogonality 114
3.3 Orthonormal Systems 123
3.4 Fourier Series 138
Chapter 4 Linear Operators. Fundamental Theorems 145
4.1 Bounded Linear Operators and Functionals 145
4.2 Spaces of Bounded Linear Operators and Dual Spaces 162
4.3 Banach-Steinhaus Theorem 173
4.4 Inverses of Operators. Banach's Theorem 180
4.5 Hahn-Banach Theorem 190
4.6 Strong and Weak Convergence 203
Chapter 5 Linear Operators on Hilbert Spaces 215
5.1 Adjoint Operators. Lax-Milgram Theorem 215
5.2 Spectral Theorem for Self-adjoint Compact Operators 229
Chapter 6 Differential Calculus in Normed Linear Spaces 257
6.1 Gateaux and Frechet Derivatives 257
6.2 Taylor's Formula, Implicit and Inverse Function Theorems 270
Bibliography 279
Index 283
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