Communications in Mathematics and Applications Volume 3 (2012), Number 1, pp. 9–15 © RGN Publications http://www.rgnpublications.com Representation of Topological Algebras by Projective Limit of Fréchet Algebras Mati Abel ⋆ Abstract. It is shown that every topological Hausdorff algebra (in particular, locally pseudoconvex Hausdorff algebra) A with jointly continuous multiplication is topologically isomorphic to a dense subalgebra of the projective limit of Fréchet (respectively, locally pseudoconvex Fréchet) algebras. In case, when A is complete, A and this projective limit of Fréchet (respectively, locally pseudoconvex Fréchet) algebras are topologically isomorphic. A partly new proof for these results from [11] are given. 1. Introduction It is well-known (published in 1952 in [10, p. 17], and in [5]) that every complete locally m-convex Hausdorff algebra is topologically isomorphic to the projective limit of Banach algebras. This result has been generalized to the case of complete locally m-(k-convex) Hausdorff algebras in [2, Theorem 5]; to the case of complete locally A-convex Hausdorff algebras in [4, Theorem 2.2], and to the case of complete locally m-pseudoconvex Hausdorff algebras in [6, pp. 202– 204]. Similar representations of topological algebras (not necessarily with jointly continuous multiplication) by projective limits of topological algebras with more simple structure are considered in [3]. It is known (see [11, Theorem 1]) that every complete topological Hausdorff algebra with jointly continuous multiplication is topologically isomorphic to the projective limit of Fréchet algebras and every complete locally convex Hausdorff algebra with jointly continuous multiplication is topologically isomorphic to the projective limit of locally convex Fréchet algebras. These results in [11] are correct, but the proofs of these are not, because in the proofs there is applied a Lemma which, first of all, is not formulated suitably 2010 Mathematics Subject Classification. Primary 46H05; Secondary 46H20. Key words and phrases. Topological algebra; Locally pseudoconvex algebra; Fréchet algebra; F - seminorm; Projective limit of topological algebras. ⋆ Research is in part supported by Estonian Science Foundation grant 7320 and by Estonian Targeted Financing Project SF0180039s08.