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1、NonarchimedeanFunctionalAnalysisPeterSchneiderVersion:25.10.20051ThisbookgrewoutofacoursewhichIgaveduringthewinterterm1997/98attheUniversit¨atM¨unster.Thecoursecoveredthematerialwhichhereispresentedinthefirstthreechapters.Thefourthmoreadvancedchapterwasaddedtogivethereaderarathe

2、rcompletetourthroughalltheimportantaspectsofthetheoryoflocallyconvexvectorspacesovernonarchimedeanfields.Thereisoneseriousrestriction,though,whichseemedinevitabletomeintheinterestofaclearpresentation.Initsdeeperaspectsthetheorydependsverymuchonthefieldbeingsphericallycompleteorno

3、t.Togiveadrasticexample,ifthefieldisnotsphericallycompletethenthereexistnonzerolocallyconvexvectorspaceswhichdonothaveasinglenonzerocontinuouslinearform.Althoughmuchprogresshasbeenmadetoovercomethisproblemareallyniceandcompletetheorywhichtoalargeextentisanalogoustoclassicalfunct

4、ionalanalysiscanonlyexistoversphericallycompletefields.Ithereforeallowedmyselftorestricttothiscasewheneveraconceptualclarityresulted.AlthoughIhopethatthistextwillalsobeusefultotheexpertsasareferencemyownmotivationforgivingthatcourseandwritingthisbookwasdifferent.Ihadthereaderinmi

5、ndwhowantstouselocallyconvexvectorspacesintheapplicationsandneedsatexttoquicklygraspthistheory.Thereareseveralareas,mostlyinnumbertheorylikep-adicmodularformsanddeformationsofGaloisrepresentationsandinrepresentationtheoryofp-adicreductivegroups,inwhichonecanobserveanincreasingi

6、nterestinmethodsfromnonarchimedeanfunctionalanalysis.Bytheway,discretelyvaluedfieldslikep-adicnumberfieldsastheyoccurintheseapplicationsaresphericallycomplete.Thisisatextbookwhichisself-containedinthesensethatitrequiresonlysomebasicknowledgeinlinearalgebraandpointsettopology.Ever

7、ythingpresentediswellknown,nothingisnewororiginal.Someofthematerialinthelastchapterappearsinprintforthefirsttime,though.InthereferencesIhavelistedallthesourcesIhavedrawnupon.Atthesametimethislistshowstothereaderwhotheprotagonistsareinthisareaofmathematics.Icertainlydonotbelongto

8、thisgroup.M¨unster,May2001PeterSchneider2Listofcontent