Fuster, R.; Marquina, A. Geometric series in incomplete normed algebras. (English) Zbl 0553.46032 Am. Math. Mon. 91, 49-51 (1984). In this elementary note the authors present simple examples of incomplete normed algebras satisfying the invertibility criterion. In fact, they establish the following. Proposition: In a normed algebra A the following conditions are equivalent. (1) G(A), the group of invertible elements, is open in A. (2) There is a real number \(\delta\) with \(0<\delta \leq 1\) such that if f is in A and \(\| f-1\| <\delta\) then f is invertible. (3) There is a real number \(\delta\) with \(0<\delta \leq 1\) such that every geometric \(\delta\)-series (i.e. the series \(\sum^{\infty}_{n=0}f^ n\) with \(f\in A\) and \(\| f\| <\delta)\) converges in A. Reviewer: N.K.Thakare Cited in 3 Documents MSC: 46H05 General theory of topological algebras Keywords:geometric series; incomplete normed algebras; invertibility criterion PDFBibTeX XMLCite \textit{R. Fuster} and \textit{A. Marquina}, Am. Math. Mon. 91, 49--51 (1984; Zbl 0553.46032) Full Text: DOI