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Differentiability in Topological Groups

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Trata sobre la extensión de la derivada de Carathodory a espacios de dimensión infinita.
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  SOOCHOWJOURNALOFMATHEMATICS Volume22,No.1,pp.39-48,January1996  DIFFERENTIABILITYINTOPOLOGICALGROUPS  BY  ERNESTOACOSTAG. Abstract.  UsingCaratheodory'sformulationofderivativewepresent aconceptofdierentiabilityoffunctionsintopologicalgroups.Wegive anecessaryconditionforuniquenessofthederivativeandprovesome oftheelementarytheoremsofdierentialcalculus. 1.Introduction  Atthebeginingof1991theAmericanMathematicalMonthlypublished thepaper\DerivativealaCaratheodory writtenbyKuhn1]whoshowsthe advantagesofaformulationofderivativegivenbyCaratheodory3]inproving thebasicdierentiabilitytheoremsofcalculus.InApril,1994,appearedthe paper\FrechetvsCaratheodory inthesamejournal2].AcostaandDelgado showtheadvantagesofthesameformulationingeneralizingthederivativeto normedspaces.Inthispaperwewillshowthattheverysameformulation allowsustogeneralizetheconceptofderivativetofunctionsintopological groups.Infact,wewillhave,inthecaseofnoncommutativegroups,aversion ofnoncommutativedierentiability.Caratheodory'sformulationofderivativeisasfollow:Afunction  f  : R  !  R  isdierentiableat  a  2  R  ifthereexistsafunction    : R  !  R  continuousat  a  andsuchthat  f  (  x  )  ;  f  (  a  )=    (  x  )(  x  ;  a  )  : ReceivedFebruary23,1995.AMSSubjectClassication.22A10.IwanttothankLorenzoAcosta,OswaldoLezama,RodrigodeCastroandMarioZu- luagafortheusefuldiscusionswehadaboutthisbeautifulsubject. 39   40ERNESTOACOSTAG. Ifsuchafunctionexists,thederivativeof  f  at  a  is    (  a  ).(Thefunctions  f  and    couldbedenedinaneighborhoodof  a  ).Itiseasytoseethatthisformulationisequivalenttotheusualone.Ob- servethatwedonotusethedierencequotientandwedonothavetoworry aboutdividing.Thisfactallowsustogeneralizederivativestofunctionsin normedspacesasfollows:Let  E  and  F  benormedspacesandlet  L  (  EF  )bethenormedspaceof linearcontinuousfunctionsfrom  E  to  F  .Afunction  f  : E  !  F  issaidtobe dierentiableat  a  2  E  ifthereexistsafunction    : E  !L  (  EF  )continuous at  a  2  E  suchthat  f  (  x  )  ;  f  (  a  )=    (  x  )(  x  ;  a  )  : Ifsuchafunctionexiststhederivativeof  f  at  a  is    (  a  ).In2]itwasshownthat    (  a  )isuniqueandthatitcoincideswithFrechet derivative.Thesteptogroupsisgivenbyobservingthat    (  a  )isahomomorphismof theadditivegroupstructuresof  E  and  F  .Let  G  and  H  betwotopological groupsandlet  Hom  (  GH  )bethespaceofcontinuoushomomorphismswith agiventopology,thenwehavethefollowingdenition: Denition1.1.  Afunction  f  : G  !  H  issaidtobedierentiableat  a  2  G  ifthereexistsafunction    : G  !  Hom  (  GH  )continuousat  a  andsuch that  f  (  x  )  f  (  a  )  ;  1  =    (  x  )  xa  ;  1  ] : (1  : 1) Ifsuchafunctionexists(and    (  a  )isunique)thederivativeof  f  at  a  is    (  a  ).Actuallywecouldwrite(1.1)infourdierentwaysif  G  and  H  arenoncom- mutativegroups: f  (  x  )  f  (  a  )  ;  1  =    (  x  )  xa  ;  1  ] f  (  x  )  f  (  a  )  ;  1  =    (  x  )  a  ;  1  x  ] f  (  a  )  ;  1  f  (  x  )=    (  x  )  xa  ;  1  ] f  (  a  )  ;  1  f  (  x  )=    (  x  )  a  ;  1  x  ] :  DIFFERENTIABILITYINTOPOLOGICALGROUPS41  Thesefourequationsdeneright-right,right-left,left-rightandleft-leftdier- entiationrespectively.Wewillseelaterthatthereisarelationshipbetween allofthem.Wegivesomepreliminariesontopologicalgroupsin  x  2.In  x  3westate somepropertiesofthefunctions    thatwecallslopefunctionsof  f  at  a  ,in  x  4 westudytheuniquenessofthederivativeandin  x  5wegivesomeexamples. 2.Preliminaries  Indenition(1.1)slopefunctionstakevaluesinthespace  Hom  (  GH  )of continuoushomomorphismsfrom  G  to  H  .Wewillchangethisspacebecause itisnotagroupandthiscausesproblemswhenwewanttocomputesecond derivatives.Considerthegroup  F  c  (  GH  )ofallcontinuousfunctionsfrom  G  to  H  andlet  Ghom  (  GH  )bethesubgroupgeneratedby  Hom  (  GH  ).Wewill callghomomorphismstheelementsof  Ghom  (  GH  ).Itisnotdiculttoshow thatif  G  and  H  arelocallycompactthen  Ghom  (  GH  )withthecompact-open topologyisatopologicalgroup.Thecompact-opentopologyin  F  c  (  GH  )isdenedasfollows:givena compactset  F    G  andopenset  O    H  wedenoteby(  FO  )thesetofall maps  h  2F  c  (  GH  )suchthat  h  (  F  )    O  .Thefamilyofsubsets(  FO  )of  F  c  (  GH  )formsabaseforthecompact-opentopologyin  F  c  (  GH  ).Withthistopologywecanproveveryeasilythatdierentiabilityimplies continuityandthattheChainRuleholds.Inordertoshowthesefactswe statethecontinuityofevaluationandcompositionofghomomorphisms. Theorem2.1.  Let  G  and  H  betwolocallycompactgroupsandlet  Ghom  (  GH  )  bethegroupofcontinuousghomomorphismswiththecompact- opentopology.Thenthemap  E  : Ghom  (  GH  )    G  !  H denedby  E  (  hx  )=  h  (  x  )  ,iscontinuous.  Proof.  Let  h  0  2  Ghom  (  GH  ), x  0  2  G  andlet  V  beaneighborhoodof   42ERNESTOACOSTAG. h  0  (  x  0  )in  H  .Since  h  0  iscontinuousand  G  islocallycompact,thereexistsa precompactneighborhood  W  of  x  0  ,suchthat  h  (   W  )    V  .Thenifwetakethe neighborhood(   WV  )    W  of(  h  0  x  0  )in  Ghom  (  GH  )    G  wehavethat  E  ((   WV  )    W  )=(   WV  )(  W  )    V: ThereforeEiscontinuousat(  h  0  x  0  ). Theorem2.2.  Let  G  ,  H  and  K  belocallycompacttopologicalgroups andconsiderthegroups  Ghom  (  GK  )  ,  Ghom  (  GH  )  and  Ghom  (  HK  )  with thecompact-opentopology.Thenthemap  C  : Ghom  (  HK  )    Ghom  (  GH  )  !  Ghom  (  GK  )  denedby  C  (  hg  )=  h    g  iscontinuous.  Proof.  Let  h  0  2  Ghom  (  HK  )and  g  0  2  Ghom  (  GH  )andlet(  FV  )be aneighborhoodof  h  0    g  0  in  Ghom  (  GK  ).Since  h  0  iscontinuous,foreach  x  2  g  0  (  F  )thereisaprecompactneighborhood  U  x  of  x  suchthat  h  0  (  U  x  )    V  .Since  g  0  (  F  )iscompactthereisanitenumberof  U  x  'sthatcover  g  0  (  F  ).Let  U  betheunionofsuch  U  x  's,then,ifwetaketheneighborhood(   UV  )    (  FU  ) of(  h  0  g  0  )in  Ghom  (  HK  )    Ghom  (  GH  )wehavethat  C  ((   UV  )    (  FU  ))=(   UV  )    (  FU  )    (  FV  ) andso  C  iscontinuousat(  h  0  g  0  ).Nowinordertoguaranteeuniquenessofthederivative,inthecasewhen  Ghom  (  GH  )iscommutative,weimposetwosucientconditionsonthetopo- logicalgroups:U1.Thereisaninteger  m  suchthattheequation  x  m  =  g  hasaunique solutionforeach  g  2  G  .U2.Foreach  x  2  G  ,thesequence  x  1  =m  n  tendsto  e  as  n  goestoinnity.Westatenowatechnicallemmaforsuchgroups.

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Jul 23, 2017
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