By Edwin Hewitt, Kenneth A. Ross

Contents: Preliminaries. - parts of the speculation of topolo- gical teams. -Integration on in the neighborhood compact areas. - In- version functionals. - Convolutions and crew representa- tions. Characters and duality of in the neighborhood 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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1, where AO=E and Al~A A ... A (ltimes). [lf rx=max {I anl. 30 Chapter 11. 2l, ... , Iannl}, then it is easy to see that no entry of the matrix Al has m absolute value exceeding (noct Hence each entry of the matrix L ~ Ai I~O . ] I~O . It is easy to show the following. (1) If Bisanymatrixin @ ~(n,K), thenexp (B-I AB) = B-I. exp (A) . B. (2) If Xl' ... , Xn are the eigenvalues of A, then exp (Xl), ... , exp (X n ) are the eigenvalues of exp (A). ] (3) The determinant of exp (A) is exp (tr A). (4) For every A, exp(A)E@~(n, K).

23) Infinite Abelian groups. (a) Every infinite Abelian group G admits a nondiserete Hausdorff topology under whieh it is a topological group. Sinee T is divisible. • eharacters of G] to distinguish an arbitrary x =f= e from e. The weakest topology for Gunder which all eharacters are eontinuous is then a nondiserete Hausdorff topology for Gunder which G is a topologieal group. 14). ) Every infinite Abelian group G admits a nondiserete Hausdorff topology under whieh there is a eountable open basis at e.

Hence there are points Xl' ... , Xn in uo(AnH) such that n U V"kXk=>Uo(AnH). k=l (2) C61 Let V be any symmetrie neighborhood of e such that V C V"k) n U. Clearly {vuoH:vEV} is a neighborhood of uoH=xoH in GJH,and thus there is a ßoEE such that ß"Zßo implies xpHE{vuoH:VE V}. Let yEE be such that Y"Zßo and y> ß"k for k=1, ... , n. Then xl'H=vuoH for some VE V; that is, xl'=vuoh for some hEH. Now h=UÖIV-IXl'EUVUC (u)3cA and, consequently, uohEuo(AnH). Applying {2}, we have UohEV"kXk for some k (k=1, ...