Polynomial Approximation and a Question of G. E. Shilov

Abstract

ABSTRACT Let E be an infinite dimensional real or complex Banach space. For n = 0,1,2,…,∞, let a n (E) be the algebra generated by all continuous polynomials on E which are homogeneous of degree ≤ n. We discuss the completion of a n (E) with respect to several natural topologies, in the real and complex case. In particular, we prove that when E is a complex Banach space whose dual has the approximation property, then the τ ω — completion of a 1 (E) can be identified with those holomorphic functions f : E → ¢ whose derivative df : E + E' is compact.