Functional continuity of topological algebras with orthonormal bases

Abstract

(i) Every complex [Formula: see text]-algebra with an identity and with an orthonormal basis is functionally continuous; (ii) Every complex complete LMC algebra with an orthogonal basis is functionally continuous; (iii) Every complex sequentially complete locally convex algebra with an unconditional orthonormal basis and with an element [Formula: see text] for which [Formula: see text]th coefficient functional value tends to infinity as [Formula: see text] tends to infinity is functionally continuous. These results are proved and an example is provided for non-extendability of these results. A representation for positive linear functionals on a sequentially complete locally convex algebra with an unconditional orthonormal basis, with an identity, and with an element [Formula: see text] mentioned in (iii) is obtained. All results are obtained only for commutative algebras.