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Geometric series in incomplete normed algebras. (English) Zbl 0553.46032

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

MSC:

46H05 General theory of topological algebras
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