When we acce pted th ekindinvitationof Prof. Dr. F. K. Scnxmrrto write a
monographon abstract harmonic analysis for the Grundlehren. der
Maihemaiischen Wissenscha/ten series, weintendedto writeall that
wecouldfindoutaboutthesubjectin a textof about 600printedpages. We
intended thatour book should be accessi ble tobeginners, and we hoped to
makeit usefulto specialists as well. These aims proved to be mutually
inconsistent. Hencethe presentvolume comprises onl y half of
theprojectedwork. Itgives all ofthe structure oftopological groups
neededfor harmonic analysisas it is known to u s; it treats integration
on locallycompact groups in detail;it contains an introductionto the
theory of group representati ons. In the second volume we will treat
harmonicanalysisoncompactgroupsand locallycompactAbeliangroups, in
considerable et d ail. Thebook is basedon courses given by E. HEWITT at
the University of Washington and the University of Uppsala,
althoughnaturallythe material of these courses has been en ormously
expanded to meet the needsof a formal monograph. Like the. other
treatments of harmonic analysisthathaveappeared since 1940, the book is
a linealdescendant of A. WEIL'S fundamentaltreatise (WElL [4J)1. The
debtof all workers in the field to WEIL'S work is wellknown and
enormous. We havealso borrowed freely from LOOMIS'S treatmentof the
subject (Lool\IIS[2 J), from NAIMARK [1J, and most especially from
PONTRYA GIN [7]. In our exposition ofthestructur e of locally compact
Abelian groups and of the PONTRYA GIN-VA N KAM PEN dualitytheorem,
wehave beenstrongly influenced byPONTRYA GIN'S treatment. We hope to
havejustified the writing of yet anothertreatiseon
abstractharmonicanalysis by taking up recentwork, by
writingoutthedetailsofeveryimportantconstruction andtheorem, andby
including a largenumberof concrete ex amplesand factsnotavailablein
other textbooks.
