By F. Brackx, Richard Delanghe, F. Sommen

ISBN-10: 0273085352

ISBN-13: 9780273085355

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It should also be noted that the same algebra plays a basic role in the paper [ABS] on Ktheory by Atiyah-Bott-Shapiro. The fact that A becomes a trace algebra in the sense of Bonsall-Goldie is revealed to be of a fundamental significance, since it allows us to pass freely from A-linear functionals to real linear functionals and vice versa. By means of this relationship we can take over without much difficulty basic principles from the general theory of locally convex spaces and its appllcations to function and distribution spaces.

Furthermore let (Ta)aE]0, 1] be an s-bounded family in v( 1 )(n;A) such that Ta s-converges to T0 if a~ 0+. Then there exist B E ~r and a family of A-valued functions (fa,fo :a E ]0,1]), all of them being continuous in a neighbourhood Wof V, such that fa converges uniformly to f 0 on Vand Ta = a8f a , To = a8f o in V. (2) If moreover (Ta>aE]0, 1] depends s-continuously upon a, then fa may be chosen in such a way that the function f given by f(a,x) = f a (x), f(O,x) = f 0 (x) is continuous in [0,1] x W.

XEK f E c0 ( Q; A) • Then P = {pK : K c n compact} induces the topology of uniform compact convergence on c( 1)(n;A). If (Ks : s = 1,2, ••• ) is the compact exhaustion of n determined by Ks = {x E Q: lxl :;; sand d(x,ll{~) ~ then Pis equivalent toP*= {pK s -}l, s 1,2, ••• , : s = 1,2, ••• }. Moreover P is also equivalent to the system of seminorms defined in a classical way on C0 (n;A) = rr C0 (n;JR). LetT E Ca*(l )(Q;A); then there exist AEPN s ~ 1 and C > 0 such that Il 0 < CpKs(f), f E c( 1)(n;A).

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Clifford Analysis by F. Brackx, Richard Delanghe, F. Sommen

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