LOCALLY PSEUDOCONVEX INDUCTIVE LIMIT 3 and E = limEα −→ is endowed with the final topology or the inductive limit topology τlimEα −→ defined by the canonical maps fα, i.e. τlimEα −→ = {U E : −1(U) τα for every α I}. Then all canonical maps : E I) are continuous homomorphisms (see [15, p. 113]). We shall consider now inductive limits of locally pseudoconvex algebras Eα. Since the inductive limit topology on E is not necessarily locally pseudoconvex, the final locally pseudoconvex topology τ is defined on E by giving the base of neighborhoods at x E on the form Lx = {x + U : U is absolutely pseudoconvex in E and −1(U) Nτα } where Nτα denotes the set of all neighborhoods of zero in Eα. Hence, (E, τ) is a locally pseudoconvex algebra, because (here θ denotes the zero element in E) is a base of absolutely pseudoconvex neighborhoods of zero in E. Herewith, the topology τ is the finest locally pseudoconvex topology on E such that is continuous for every α I. Later on we shall say in this case that τ is the locally pseudoconvex inductive limit topology on the inductive limit of locally pseudoconvex algebras. In case, when E = limEα, −→ where every is a subset of E, E = α∈I and for every α, β I there exists γ I such that and Eγ, then the notation −→ instead of limEα −→ is used. Next we recall several classes of topological algebras introduced in [4]. A topo- logical algebra (E, τ) is a generalized LF-algebra (in short, LFg-algebra), if E is an inductive limit of F -algebras1 (Eα,τα) such that (2) E = −→ (as sets), the topology τ on E coincides with the inductive limit topology, defined by canonical maps, and the topology, induced on every by τ, coincides with the original topology τα on Eα. Moreover, (E, τ) is an LF-algebra, if E is an inductive limit of an increasing sequence of F -algebras (En,τn) such that holds (3) E = −→ En (as sets), the topology τ coincides with the inductive limit topology on E, defined by canonical maps, and for each n N the topology induced on En by En+1 and by the topology τ are identical to the original topology τn on En. In particular case, when every in (2) (En in (3)) is a locally pseudoconvex F -algebra and τ on E coincides with the locally pseudoconvex inductive limit topology, defined by canon- ical maps, then LFpg-algebra (LFp-algebra) instead of LFg-algebra (respectively LF-algebra) is used. Moreover, when all (respectively, En) and E are locally k-convex algebras for some k (0, 1], then the term k-LFg-algebra (respectively, k-LF-algebra) are used. 1An F -algebra is a metrizable and complete algebra.
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