Mathematics

# Analytically Uniform Spaces and their Applications to by Carlos A Berenstein By Carlos A Berenstein

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N (I~(¢)[e ~(~)) (~) I[1¢111(~) = sup C = ~+i~ ~ ~n. ]~n). the system of norms { III"IIIl,s (m) }A,s >0' (~) where (rl¢ in (1) is equivalent { If'If ~ ) }~ > 0 two systems function H K as HK(r~ ) (I'l(~)}l> 0 (kY}k> 1 III below. If K is a compact set in ~n, the supporting K is Thus, we obtain where B is the sequence in Chapter test ~ ~ ~Z, for some special choice of ~. by taking the Gevrey classes This \$ of any Beurling There are some other function also be obtained as ~ 2. transform is an entire function.

A3o I~(~)I for J K Jo" since ! In the remaining strips, we have ~ ~ i. C~exp(-6m(~) r s AI U . . U Ajo_l. Hence Therefore, by choosing ~ ~ X + Aajo + Alql) ! C6exp(-~(~)) (CC;I)~ s ~(C,X,{rj},{aj}). c. topology ~(~) on ~ 32 having for the basis of neighborhoods of the origin the system of all sets ~ of the form (5). Topology % ( % ) . Let {Hs}s>l be any concave sequence of positive Hs/S ÷ 0. Fix a positive number ~ and a bounded numbers, H s ~ ~, sequence (~s}s>l of positive numbers. Then the series oo {6) k(~) = k((Hs};(Cs};p;~ ) = [ s=l ~seXp[-(s+p)oJ(~) + Hslrll] is locally uniformly convergent in ~n and defines a majorant in the sense of Chap.

52 n j=l ~jk (~oj) Therefore, ly ~ _< if ~. denotes 3 compatible A(m,c) ~(D), 1 _< j _< n} must be a defined as the unique bornological with a fundamental = {~ ~ D : : mj ~ J(j, the fami- n Let D be the vector space @ ~ equipped ~ j=l j for ~' ~" with the topology . the family ~f(Zc) for the space ~ . mn(¢n) BAU-structure n < n ~ ~j X (~j) -j =I ~k(~) sup topology system of bounded sets of the form (~(~)I/m(~)) < c}. c. ~ of D to a subspace is the whole space ~ From the discussion is an iU-structure Def.