Structure of topological groups, integration theory, group representations
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Structure of topological groups, integration theory, group representations
（Die Grundlehren der mathematischen Wissenschaften, 115 . Abstract Harmonic analysis ; v. 1）
Springer-Verlag, 1979
2d ed
- : N.Y.
- : Berlin
- : N.Y. pbk.
- : Berlin pbk.
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Includes indexes
Description and Table of Contents
- Volume
-
: N.Y. pbk. ISBN 9780387941905
Description
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.
Table of Contents
One: Preliminaries.- Section 1. Notationand terminology.- Section 2. Group theory.- Section 3. Topology.- ChapterTwo : Elementsof thetheoryof topological groups.- Section 4. Basic definitions and facts.- Section 5. Subgroups and quotient groups.- Section 6. Product groups and projective limits.- Section 7. Properties of topologicalgroups involving connectedness.- Section 8. Invariant pseudo-metrics and separation axioms.- Section 9. Structure theory for compact and locally compact Abelian groups.- Section 10. Some special locally compact Abelian groups.- Three: Integration on locally compact spaces.- Section 11. Extension of a linear functional and construction of a measure.- Section 12. The spaces Lp(X) (1 ? p ? ?).- Section 13. Integration on product spaces.- Section 14. Complex measures.- Four: Invariant functionals.- Section 15. The Haar integral.- Section 16. More about Haar measure.- Section 17. Invariant means defined for all bounded functions.- Section 18. Invariant means on almost periodic functions.- Five: Convolutions and group representations.- Section 19. Introduction to convolutions.- Section 20. Convolutions of functions and measures.- Section 21. Introduction to representation theory.- Section 22. Unitary representations of locally compact groups.- Six : Characters and duality of locally compact Abelian groups.- Section 23. The character group of a locally compact Abelian group.- Section 24. The duality theorem.- Section 25. Special structure theorems.- Section 26. Miscellaneous consequences of the duality theorem.- Appendix A: Abelian groups.- B: Topological linear spaces.- C: Introduction to normed algebras.- Index of symbols.- Index of authors and terms.
- Volume
-
: Berlin ISBN 9783540094340
Description
Contents: Preliminaries. - Elements of the theory of topolo- gical groups. -Integration on locally compact spaces. - In- variant functionals. - Convolutions and group representa- tions. Characters and duality of locally compact Abelian groups. - Appendix: Abelian groups. Topological linear spa- ces. Introduction to normed algebras. - Bibliography. - In- dex of symbols. - Index of authors and terms.
- Volume
-
: Berlin pbk. ISBN 9783540941903
Description
The book is based on courses given by E. Hewitt at the University of Washington and the University of Uppsala. The book is intended to be readable by students who have had basic graduate courses in real analysis, set-theoretic topology, and algebra. That is, the reader should know elementary set theory, set-theoretic topology, measure theory, and algebra. The book begins with preliminaries in notation and terminology, group theory, and topology. It continues with elements of the theory of topological groups, the integration on locally compact spaces, and invariant functionals. The book concludes with convolutions and group representations, and characters and duality of locally compact Abelian groups.
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