For a Hausdorff locally convex space E with topological dual E', the following properties are equivalent:(B 1) Every bound linear transformation from P into any locally convex space is continuous; (B 2) No strictly stronger locally convex topology on E has the same bound sets; (B 3) Every bornivore set is a neighborhood of zero (a bornivore set is a convex, equilibrated set absorbing every bound set); (B 4) E is the (linear) inductive limit [7, pp. 61-66] of normed spaces {Ea} with respect to linear maps ga:Ea-*P such that U" ga(Ea) =P; (B 5) The topology of P is r(E, E') (the strongest locally convex topology on P yielding E' as topological dual), and every bound linear form on P is continuous. ( . Our purpose here is to formulate the analogous notion of the algebraic inductive limit of locally m-convex algebras and to study those locally mconvex algebras which are algebraic inductive limits of normed algebras.In §1 we summarize briefly the basic definitions and some of the elementary results in the theory of locally w-convex algebras. In §2, after defining the notion of the algebraic inductive limit of locally m-convex algebras in an obvious way, we consider the following problem. If E is an algebra, \Ea\ a family of locally w-convex algebras, ga:P"->P a homomorphism for all a, there exist on P both the algebraic inductive limit topology with respect to locally w-convex algebras Ea and homomorphisms ga, and the linear inductive limit topology with respect to locally convex spaces Ea and linear maps ga; when do they coincide ? This question is important, for the topologies of certain important locally convex spaces (occurring, for example, in the theory of distributions and the theory of integration) are defined as linear inductive limits of locally convex spaces which are also locally wz-convex algebras, with respect to linear maps which are also homomorphisms of the algebras considered. To establish that such topologies are actually locally w-convex, and thus fall within the purview of topological algebra, it is desirable to have general criteria insuring that the two inductive limit topologies coincide.