Division mit Rest in Banach-Algebren konvergenter Potenzreihen

Abstract

In [1, Kapitel I] Grauert and Remmert prove the Weierstras division theorem for convergent power series by use of the Banach algebra of all power series converging absolutely on the closure $\overline {P}$ of an open polycylinder P. An analogous argument is given by Hormander in [3, Section 6.1] who, however, uses the Banach algebra of all functions holomorphic and bounded on P. ¶The present paper gives an axiomatic approach to this Weierstras division in Banach algebras of convergent power series which is based merely on a simple application of the geometric series and which works for both types of Banach algebras mentioned above and also, e.g, for the Banach algebra of all functions which are holomorphic on P and continuous on $\overline {P}$ .