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By Gawronski W., Shawyer B. L., Trautner R.

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Additional resources for A Banach space version of Okada's theorem on summability of power series

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B) A is continuous at one point of c) There exists a neighborhood such that A is bounded in V Em. 7 If P E pa(E;F) the following are equivalent: a) P is continuous. b) P is continuous at one point of c) There exists a non-empty open subset that is bounded in P PROOF: E. U of E such U. The implications a) 3 b) a c) are easily verified. We prove the implication c) a a). By c), there exists a non-empty open subset M and a constant 1) I\P(X)~ Since t: E -t E 2 s M U 0 of such that for every x E U. is non-empty there exists be the translation: 1) is equivalent to U t(x) = x-x 0 ) x 0 E U.

Is a non-empty open If g: V -I V is defined it is easy to see that g E #(V;F). {A E C : then E U] 0 , s+Xx 33 0 1x1 s I; p} c V and s o , by a), we have f o r every m E N. The Taylor series o f at f 5 , C P&(Z-5;), con- &=O f(z) verges uniformly t o With x f 0, choose E in a ball E IR, 0 < 0 Bu(5;), < p, u E iR, u 7 0. sufficiently small m so that the series C P,(Z-T) converges uniformly to f(z) &=O in the closed ball with centre 5 and radius ~llxll. Then if Therefore, from (*), we have Since the series converges uniformly in this yields {hEC; IXI=c], 4 CHAPTER 34 x = 0, the proposition is trivial.

The implications a) 3 b) a c) are easily verified. We prove the implication c) a a). By c), there exists a non-empty open subset M and a constant 1) I\P(X)~ Since t: E -t E 2 s M U 0 of such that for every x E U. is non-empty there exists be the translation: 1) is equivalent to U t(x) = x-x 0 ) x 0 E U. x E E. Let Then E NOTATION AND TERMINOLOGY. 6, that ll~(~)ll s M Q = t is a is a continuous y E V. P = Qot, and Q. ,m, and bounded in a subset if Since V and is a polynomial, and, by 2), equivalent to continuity of continuous.

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A Banach space version of Okada's theorem on summability of power series by Gawronski W., Shawyer B. L., Trautner R.


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