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1、MathematicalSurveysandMonographsVolume53TheConvenientSettingofGlobalAnalysisAndreasKrieglPeterW.MichorATHEMATICMALNTRHTOSMHEISITWSAOICCIREAmericanMathematicalSocietyEAGEWMETMYAF8O8UNDED18EditorialBoardHowardA.MasurMichaelRenardyTudorStefanRatiu,Chair1991MathematicsSubjectClassification.Primary22E
2、65,26E15,26E20,46A17,46G05,46G20,46E25,46E50,58B10,58B12,58B20,58B25,58C20,46E50,58D05,58D10,58D15,58D17,58D19,58F25;Secondary22E45,58C40,22E67,46A16,57N20,58B05,58D07,58D25,58D27,58F05,58F06,58F07.Abstract.Theaimofthisbookistolayfoundationsofdifferentialcalculusininfinitedimensionsandtodiscusstho
3、seapplicationsininfinitedimensionaldifferentialgeometryandglobalanalysiswhichdonotinvolveSobolevcompletionsandfixedpointtheory.Theapproachisverysimple:Amappingiscalledsmoothifitmapssmoothcurvestosmoothcurves.Allotherpropertiesareprovedresultsandnotassumptions:Likechainrule,existenceandlinearityofde
4、rivatives,powerfulsmoothuniformlyboundednesstheoremsareavailable.UptoFr´echetspacesthisnotionofsmoothnesscoincideswithallknownreasonableconcepts.Inthesamespiritcalculusofholo-morphicmappings(includingHartogs’theoremandholomorphicuniformboundednesstheorems)andcalculusofrealanalyticmappingsaredeve
5、loped.Existenceofsmoothpartitionsofunity,thefoundationsofmanifoldtheoryininfinitedimensions,therelationbetweentangentvectorsandderivations,anddifferentialformsarediscussedthoroughly.SpecialemphasisisgiventothenotionofregularinfinitedimensionalLiegroups.Manyapplicationsofthistheoryareincluded:manifo
6、ldsofsmoothmappings,groupsofdiffeomorphisms,geodesicsonspacesofRiemannianmetrics,directlimitmanifolds,perturbationtheoryofoperators,anddifferentiabilityquestionsofinfinitedimensionalrepresentations.CorrectionsandcomplementstothisbookwillbepostedontheinternetattheURLhttp://www.mat.univie.ac.at/~mich
7、or/apbook.psLibraryofCongressCataloging-in-PublicationDataKriegl,Andreas.Theconvenientsettingofglobalanalysis/AndreasKriegl,PeterW.Michor.p.cm.—(Mathematicalsurveysandmonographs,ISSN0076-5376;v.53)Includesbibliographicalrefe
