By Luigi Prouse, Giovanni, Amerio

ISBN-10: 1475712561

ISBN-13: 9781475712568

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**Example text**

3) it follows, in fact, that a(A; (x*,f(t)) = A(x*,f(t)e- illt ») = (x*,a(A) and, consequently, a(A; (x*,J(t)) =0 for A¢ {An}. Furthermore, (x*, Qm(f(t))) = i" JLmk(X*,a(Ak)e illkt , 1 that is, (x*, Qm(f(t))) is the mth Bochner polynomial constructed from the basis B and corresponding to the function (x*,f(t); this is also true if, for some k, (x*,a(Ak)=O. 8) is proved. P. VIII. a(A)=O => 45 f(t)=O (uniqueness theorem). 7), Qm(f(t)) =0, "1m. 8) itfollows that f(t) =0. Finally, let us extend Bochner's criterion.

Furthermore, let (t EJ) x = f(t) be a function with values in X. , 'r/x* E X*. p. e. such that

M that is, {'} •. , with centers at the points 2nm7T. } which -+ + 00 when n -+ + 00. }. ), Hence, n {'}. }. d. p. A similar example can be given for X =co, space of continuous numerical functions on the interval Of----l1. The problem now arises of determining whether there are some particular Banach spaces to which the theorem of BoW-Bohr can be extended word by word. This actually can be done for notable spaces-for example, for INTRODUCTION AND STATEMENTS 55 Hilbert spaces and, more generally, for uniformly convex spaces (Amerio [4], [9]).