An Infinite Dimensional Version of a Theorem of Bernstein
نویسنده
چکیده
1. Introduction. Let (R") is the algebra Q1(Rn) of all real-valued functions of class C1. In other words, for every fEG1(Rn) there is a sequence {pn} of polynomials such that pn—*/ uniformly on bounded sets and / " ' —»/' uniformly on bounded sets. In this paper we define the algebra (P(X) of polynomials in a Banach space X and determine its closure for a restricted class of reflexive Banach spaces (Theorem 8). Thus, Theorem 8 answers a question raised by the author in [2]. In what follows, weak convergence in X will be denoted by xn—^x and strong convergence by xB—>x. Henceforth we will assume all Banach spaces to be separable.
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