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The linear functionals on fundamental locally multiplicative topological algebras

In this paper we study the dual space of fundamental locally multiplicative topological algebras and prove some results for linear and multiplicative linear functionals on these algebras. An investigation on locally compactness of the carrier space of these algebras is the last part of this note. | Turk J Math 34 (2010) , 385 – 391. ¨ ITAK ˙ c TUB doi:10.3906/mat-0810-12 The linear functionals on fundamental locally multiplicative topological algebras E. Ansari-Piri Abstract In this paper we study the dual space of fundamental locally multiplicative topological algebras and prove some results for linear and multiplicative linear functionals on these algebras. An investigation on locally compactness of the carrier space of these algebras is the last part of this note. Key word and phrases: Multiplicative linear functionals, carrier space, fundamental locally multiplicative topological algebras. 1. Introduction In [2] we have extended and proved the famous Cohen factorization theorem for complete metrizable fundamental topological algebras, where the meaning of fundamental topological algebras generalizing both local boundedness and local convexity is initially introduced in [1] and [2]. Yet, some of the basic theorems are proved on fundamental topological vector spaces in [3]. To answer the wide question, which properties of the well-known topological algebras can be extended to fundamental topological algebras, we have introduced in [5] the notion of fundamental locally multiplicative topological algebra (abbreviated by FLM ) with a property very similar to the normed algebras. In this note, we give an example of an FLM algebra which is neither locally bounded nor locally convex and in [6], by discussing on a necessary condition for a fundamental topological algebra to be FLM, we gave an example of a complete metrizable separable locally multiplicative convex topological algebra which is not FLM. In section 2, we have gathered a collection of definitions and related results, and in section 3 we present a discussion on the relation between FLM and locally bounded algebras. In section 4, we define a norm on a subspace of the algebraic dual space of an FLM algebra where we show that the multiplicative linear functionals have a norm no greater than one. In .