《泛函分析(英文版)》在Princeton使用,同时在其它学校,比如UCLA等名校也在本科生教学中得到使用。其教学目的是,用统一的、联系的观点来把现代分析的“核心”内容教给本科生,力图使本科生的分析学课程能接上现代数学研究的脉络。
; Z3 X" _ V1 C 目录
" X5 @- S( m( F- }4 ?5 J. e6 d Foreword1 I: S+ e6 x% {6 k* \
Preface$ |- C; C, W) K4 Y" I* U
Chapter 1.Lp Spaces and Banach Spaces6 @6 ^/ _) \ H7 n: N
1 Lp spaces& J1 O" Z% f9 T' D$ q% l X- w
1.1 The Holder and Minkowski inequalities1.2 Corupleteness of Lp( p8 U+ d% }, \
1.3 Further remarks
" e0 x1 e9 Q* J0 E6 ^1 r- s B 2 The case p = ∞2 c0 T. @% s8 I% y1 \! w
3 Banach spaces
1 i m! x, {4 W8 o 3.1 Examples
: P0 @& Y) L" ~5 \) d+ d 3.2 Linear functionals and the dual of a Banach space4 The dual space of Lp when l < p < ∞ _8 T# s2 l3 Q% S6 W4 R+ w+ u
5 More about linear functionals6 J- D& z& O0 f E5 k9 o6 A3 `
5.1 Separation of convex sets
/ @. ]3 B* @2 Z& Q 5.2 The Hahn-Banach Theorem$ |. `3 f2 U0 r$ [1 n; h' f
5.3 Some consequences o0 l$ h& s; d
5.4 The Droblem of measure/ {# s, B% D# P; f
6 Complex Lp and Banach spaces
- S& M+ Y5 S, R, X7 O& B 7 Appendix: The dual of C(X)7 o0 C: U( o8 [6 N, J6 o5 e
7.1 The case of positive linear functionals
5 e" H/ M0 f- B, c 7.2 The main result
u$ b; B# H* ]! _& i8 S 7.3 An extension
- \) j: n. |, E/ _9 z% } `/ } 8 Exercises1 |- r6 z7 n2 W M( {
9 Problems
, c' p4 B# B; i+ ] Chapter 2.Lp Spaces in Harmonic Analysis
8 H8 O- a6 d6 @3 \. y$ \ 1 Early Motivations* C; t* W6 ~+ a1 D5 I; [- v' [ r
2 The Riesz interpolation theorem
7 j [8 M1 L' c 2.1 Some examples" ~ K( q; e4 g0 ^
3 The Lp t.heory of the Hilbert t.ransform- @" X f. F6 @+ S' X: h6 p. e
3.1 The L2 formalism; T2 |7 @1 D1 x7 O. {9 L
3.2 The Lp theorem, M$ B& D; d, z0 }" M
3.3 Proof of Theorem 3.2# B; \: B8 z" k, c$ g7 G
4 The maximal function and weak-type estimates1 r% p: ]% [9 ~6 f- y+ _4 s# c
4.1 The Lp inequality2 }- r r& w3 N; k
……
) \# N1 C1 K1 Z3 n* {% s Chapter 3.Distributions: Generalized FunctionsChapter 4.Applications of the Baire Category TheoremChapter 5.Rudiments of Probability TheoryChapter 6.An Introduction to Brownian MotionChapter 7.A Glimpse into Several Complex VariablesChapter 8.Oscillatory Integrals in Fourier AnalysisNotes and References" Z/ h, c2 t1 n2 j3 V1 v
Bibliography; V- S# @! J" a2 _ p. E9 u
Svmbol Glossary/ D, m9 U( O$ G8 N8 s) h) @
Index
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