Dales, H.G. and McClure, J. P. (1975) Completion of normed algebras of polunomials. Journal of the Australian Mathematical Society, 20 (4). pp. 504-510. ISSN 1446-7887
Abstract
Let P be the algebra of polynomials in one inderminate x over the complex field C. Suppose xs2225 · xs2225 is a norm on P such that the coefficient functionals cj: ∑αix1 → αj (j = 0,1,2,…) are all continuous with respect to xs2225·xs2225, and Let K xs2282 C be the set of characters on P which are xs2225·xs2225-continuous. then K is compact, C\K is connected, and 0xs2208K. K. Let A be the completion of P with respect to xs2225·xs2225. Then A is a singly generated Banach algebra, with space of characters (homeomorphic with) K. The functionals cj have unique extensions to bounded linear functionals on A, and the map a →∑Ci(a)xi (a xs2208 A) is a homomorphism from A onto an algebra of formal power series with coefficients in C. We say that A is an algebra of power series if this homomorphism is one-to-one, that is if a xs2208 A and a≠O imply cj(a)≠ 0 for some j.