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The purpose of this paper is to introduce the class of locally boundedly niultiplicatively convex algebras. This new class has been compared with other generalizations of BANACH algebras, for instance, with locally multiplicatively convex algebras introduced and studied by ARENS and MICHAEL and A-convex algebras defined and studied by COCHRAN, KEOWN & WILLIAMS. Necessary and sufficient conditions on the dual pair (E, E') are given for E to have a locally boundedly multiplicatively convex topology compatible with duality. Relations between the conditions of local bounded niultiplicative convexity of E with the strong topology and local multiplicative convexity of the bidunl with ARENS product and that of the associated bornological structure have been studied.A good number of examples have been given from Function spaces, in particular, weighted function spaces and algebras with the mixed topology.