内容简介

《希尔伯特空间及其空间导论(英文版)(第3版)》是一部学习希尔伯特空间的入门级教程。无论是学生还是科研人员,都将从《希尔伯特空间及其空间导论(英文版)(第3版)》的特别表达中受益。《希尔伯特空间及其空间导论(英文版)(第3版)》在原来版本的基础上做了不少改动,新增加了一部分讲述Sobolev空间,展开讲述了有限维赋范空间,有关小波的一章做了全面更新。并且包括了积分和微分方程、量子力学、最优化、变分和控制问题、逼近理论问题、非线性不稳定性和分岔理论的多种应用。在众多希尔伯特空间的书中,《希尔伯特空间及其空间导论(英文版)(第3版)》在讲述勒贝格积分方面独具特色。学习泛函分析和希尔伯特理论的老师和学生都十分推崇这《希尔伯特空间及其空间导论(英文版)(第3版)》作为教材或者参考书。

目录

prefacetothethirdedition
prefacetothesecondedition
prefacetothefirstedition
chapter1nermedvectorspaces
1.1introduction
1.2vectorspaces
1.3normedspaces
1.4knachspaces
1.slinearmappings
1.6contractionmappingsandthebanachfixedpointtheorem
1.7exercises

chapter2thelebesgueintegral
2.1introduction
2.2stepfunctions
2.3lebesl~eintelfablefunctions
2.4theabsolutevalueofoninteifablefunction
2.5seriesofintelqblefunctionsso
2.6norminl1(r)
2.7convergencealmosteverywheress
2.8fundamentolconvergencetheorems
2.9locallyintegmblefunctions
2.10thelebesgueintegralandtheriemannintegral
2.11lebesguemeasureonr
2.12complex-valuedlebesgueintegrablefunctions
2.13thespaceslp(r)
2.14lebesgueintegrablefunctionsonrn
2.15convolution
2.16exercises

chapter3hilbertspacesandorthonormalsystems
3.1introduction
3.2innerproductspaces
3.3hilbertspaces
3.4orthogonalandorthonormalsystems
3.5trigonometricfourierseries
3.6orthogonalcomplementsandprojections
3.7linearfunctionalsandtherieszrepresentationtheorem
3.8exercises

chapter4linearoperatorsonhilbertspaces
4.1introduction
4.2examplesofoperators
4.3bilinearfunctionalsandquadraticforms
4.4adjointandseif-adjointoperators
4.5invertible,normal,isometric,andunitaryoperators
4.6positiveoperators
4.7projectionoperators
4.8compactoperators
4.9eigenvaluesandeigenvectors
4.10spectraldecomposition
4.11unboundedoperators
4.12exercises

chapter5applicationstointegralanddifferentialequations
5.1introduction
5.2basicexistencetheorems
5.3fredholmintegralequations
5.4methodofsuccessiveapproximations
5.5volterraintegralequations
5.6methodofsolutionforaseparablekernel
5.7volterraintegralequationsofthefirstkindandabel'sintegralequation
5.8ordinarydifferentialequationsanddifferentialoperators
5.9sturm-liouvillesystems
5.10inversedifferentialoperatorsandgreen'sfunctions
5.11thefouriertransform
5.12applicationsofthefouriertransformtoordinarydifferentialequationsandintegralequations
6.13exercises

chapter6generalizedfunctionsandpartialdifferentialequations
6.1introduction
6.2distributions
6.3*sobolevspaces
6.4fundamentalsolutionsandgreen'sfunctionsforpartialdifferentialequations
6.5weaksolutionsofellipticboundaryvalueproblems
6.6examplesofapplicationsofthefouriertransformtopartialdifferentialequations
6.7exercises

chapter7mathematicalfoundationsof@uantummechanics
7.1introduction
7.2basicconceptsandequationsofclassicalmechanicspoisson'sbracketsinmechanics
7.3basicconceptsandpostulatesofquantummechanics
7.4theheisenberguncertaintyprinciple
7.5theschrodingerequationofmotion
7.6theschrodingerpicture
7.7theheisenbergpictureandtheheisenbergequationofmotion
7.8theinteractionpicture
7.9thelinearharmonicoscillator
7.10angularmomentumoperators
7.11thediracrelativisticwaveequation
7.12exercises

chapter8waveletsandwavelettransforms
8.1briefhistoricalremarks
8.2continuouswavelettransforms
8.3thediscretewavelettransform
8.4multirosolutionanalysisandorthonormalbasesofwavelets
8.5examplesoforthonormalwavelets
8.6exercises

chapter9optimizationproblemsandothermiscellaneousapplications
9.1introduction
9.2thegateauxandfrechetdifferentials
9.3optimizationproblemsandtheeuler-lagrangeequations
9.4minimizationofquadraticfunctionalss0s
9.5variationalinequalitiess07
9.6optimalcontrolproblemsfordynamicalsystems
9.7approximationtheory
9.8theshannonsamplingtheorem
9.9linearandnonlinearstability
9.10bifurcationtheory
9.11exercises
hintsandanswerstoselectedexercises
bibliography
index

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