现代数学物理方法（第1卷 英文版）

　　This book is the first of a multivolume series devoted to an exposition of functional analysis methods in modem mathematical physics. It describes the fundamental principles of functional analysis and is essentially self-contained， although there are occasional references to later volumes. We have included a few applications when we thought that they would provide motivation for the reader. Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics， modem physics， and partial differenrial equations.本书为英文版。

暂缺《现代数学物理方法（第1卷 英文版）》作者简介

Preface
Introduction
ContentsofOtherVolumes
I:PRELIMINARIES
1.Setsandfunctions
2.Metricandnormedlinearspaces
AppendixLirasupandliminf
3.TheLebesgueintegral
4.Abstractmeasuretheory
5.Twoconrergencearguments
6.Equicontinuity
Notes
Problems
II:HILBERTSPACES
1.ThegeometryofHilbertspace
2.TheRieszlemma
3.Orthonormalbases
4.TensorproductsofHilbertspaces
5.Ergodictheory:anintroduction
Notes
Problems
III:BANACHSPACES
1.Definitionandexamples
2.Dualsanddoubleduals
3.TheHahn-Banachtheorem
4.OperationsonBanachspaces
5.TheBairecategorytheoremanditsconsequences
Notes
Problems
IV:TOPOLOGICALSPACES
1.Generalnotions
2.Netsandconvergence
3.Compactness
AppendixTheStone-Weierstrasstheorem
4.Measuretheoryoncompactspaces
5.WeaktopologiesonBanachspaces
AppendixWeakandstrongmeasurability
Notes
Problems
V:LOCALLYCONVEXSPACES
1.Generalproperties
2.Frdchetspaces
3.Functionsofrapiddeceaseandthetempereddistributions
AppendixTheN-representationforand
4.Inductivelimits:generalizedfunctionsandweaksolutionsof
partialdifferentialequations
5.Fixedpointtheorems
6.Applicationsoffixedpointtheorems
7.Topologiesonlocallyconvexspaces:dualitytheoryandthe
strongdualtopology
AppendixPolarsandtheMackey-Arenstheorem
Notes
Problems
VI:BOUNDEDOPERATORS
1.Topologiesonboundedoperators
2.Adjoints
3.Thespectrum
4.Positiveoperatorsandthepolardecomposition
5.Compactoperators
6.ThetraceclassandHilbert-Schmidtideals
Notes
Problems
VII:THESPECTRALTHEOREM
1.Thecontinuousfunctionalcalculus
2.Thespectralmeasures
3.Spectralprojections
4.Ergodictheoryrevisited:Koopmanism
Notes
Problems
VIII:UNBOUNDEDOPERATORS
1.Domains,graphs,adjoints,andspectrum
2.Symmetricandself-adjointoperators:thebasiccriterion
forself-adjointness
3.Thespectraltheorem
4.Stone'stheorem
5.Formalmanipulationisatouchybusiness:Nelson's
example
6.Quadraticforms
7.Convergenceofunboundedoperators
8.TheTrotterproductformula
9.Thepolardecompositionforclosedoperators
10.Tensorproducts
11.Threemathematicalproblemsinquantummechanics
Notes
Problems
THEFOURIERTRANSFORM
1.TheFouriertransformon(Rn)and(Rn),convolutions
2.TherangeoftheFouriertransform:Classicalspaces
3.TherangeoftheFouriertransform:Analyticity
Notes
Problems
SUPPLEMENTARYMATERIAL
II.2.ApplicationsoftheRieszlemma
III.1.BasicpropertiesofLpspaces
IV.3.ProofofTychonoffstheorem
IV.4.TheRiesz-MarkovtheoremJorX=[0,1]
IV.5.MinimizationofJunctionals
V.5.Proofsofsometheoremsinnonlinearfunctionalanalysis
VI.5.Applicationsofcompactoperators
VIII.7.Monotoneconvergenceforforms
VIII.8.MoreontheTrotterproductformula
Usesofthemaximumprinciple
Notes
Problems
ListofSymbols
Index