Download Abstract Harmonic Analysis: Volume 1: Structure of by Edwin Hewitt, Kenneth A. Ross PDF

By Edwin Hewitt, Kenneth A. Ross

Contents: Preliminaries. - parts of the speculation of topolo- gical teams. -Integration on in the community compact areas. - In- version functionals. - Convolutions and crew representa- tions. Characters and duality of in the community compact Abelian teams. - Appendix: Abelian teams. Topological linear spa- ces. creation to normed algebras. - Bibliography. - In- dex of symbols. - Index of authors and phrases.

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Additional info for Abstract Harmonic Analysis: Volume 1: Structure of Topological Groups Integration Theory Group Representations

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Let cp be the natural mapping x_xH of G onto GjH. J{GjH) if and only if cp-1{{xH:XEX}) isopen in G. In other words, {xH: XEX} is openif andonlyif U {xH: XEX}=XH is open in G. Since {xH:XEX}={xH:XEXH}, it follows that every open set in GjH has the form {uH:UEU} where U is an open subset of G. J{GjH) consists of all sets of the form {uH:UE U} where U is open. It is also easy to see that sets of the form {u H: u EU}, where U is open in G and aE U, form an open basis at each point aH of GjH. 16) Theorem.

GIH) is the weakest topology, {0, GIH}, if and only if l t =G. ) Let G be a topological group and H a subgroup of G. If H and GIH have countable open bases at each point, so does G. }::"=l be a sequence of symmetrie neighborhoods of e in G such that W"\l cw,. nH}::"=l is an open basis at e for the group H. Let {{uH:UEl:T,,}}::"=l be an open basis at the point Hof GIH, where the l7,. are neighborhoods of e in G. For n = 1, 2, ... , let and Q.. = 1l n P2 n ... n P". I t is easy to see that every Q" is a neighborhood of e.

Thus ih is a homeomorphism, we have (A n W) . H = V n (AH) for some open subset V of G. 32). Suppose that 7: is a homeomorphism, that A is locally compact, and that H is closed. Then AH is closedin G. 22), AJ(A nH) is locally compact. 11). This obviously implies that AH is closed in G. J (g) Let G be a topologieal group and H anormal subgroup of G. Then GJl1" is topologically isomorphie with (GJH)J{H}-, where {H}- is the closure in GJH of the identity element H. J (h) Let G be a topological group with a compact normal subgroup H such that GJH is compactly generated.

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