LOCALLY PSEUDOCONVEX INDUCTIVE LIMIT 11 with u1,...,un U, μ1,...,μn K and ∑n ν=1 | μν |k 1. Since I is a directed set, then there is an α I such that for every ν with 1 ν n. Hence x and ρE(x) ρEα (x) n ν=1 | μν | ρEα (uν) ε n ν=1 | μν | ε n ν=1 | μν |k ε for every x Vk. So, ρE is upper semicontinuous at zero in E. Taking this into account, the set U = {x E : ρE(x) 1} is an open neighborhood of zero in E. Suppose that U QinvE. Then there is an element u U such that u QinvE. Now 1 spE(u), but this is not possible, because ρE(u) 1. Thus, U QinvE and E is a Q-algebra. To prove (5), let be for every α I an absolutely kα-convex neighborhood of zero in such that QinvEα, since QinvEα is a neighborhood of zero in Eα. Then O = Γkα0 (fα0 (Oα0 )) is a neighborhood of zero in the topology τ. Indeed, fβα0 (Oα0 ) is a neighborhood of zero in by hypothesis and fβα0 (Oα0 ) −1 [fβ fβα0 (Oα0 )] = −1 [fα0 (Oα0 )] −1 (O) for every β I. To show that O QinvE, let x O. Then x = n ν=1 μν with n N, x1,...,xn fα0 (Oα0 ), μ1, ··· , μn K and n ν=1 | μν |kα0 1. Now there exist o1,...,on Oα0 such that = fα0 (oν) for each ν. Hence x = fα0 n ν=1 μν fα0 (Γkα0 (Oα0 )) = fα0 (Oα0 ) QinvE. It means that O QinvE. Hence QinvE is a neighborhood of zero in E in the locally pseudoconvex inductive limit topology on E. Remark 1. The proof of the case 2 from (5) follows that (E, τ) is a Q-algebra does not need the restriction E = −→ to Eα. Corollary 2.5. Let (E, τ) be a LFpg-algebra defined by locally pseudoconvex F -algebras with α I and let ρEα be a seminorm on for each α I. If, in addition, every (Eα,τα) is a Q-algebra, then (E, τ) is also a Q-LFpg-algebra. Proposition 2.6. Let (E, τ) be a LFp-algebra defined by locally kn-convex F -algebras (En,τn) and let k = inf{kn : n N} 0. If, in addition, every (En,τn) is a Q-algebra, then (E, τ) is a Q-LFp-algebra. Proof. Since every (En,τn) is a Q-algebra, then QinvEn is a neighborhood of zero in En for each n N. Hence, there is an absolutely kn-convex neighborhood On of zero such that On QinvEn for each n N. We can assume that O1 O2 · · · On ··· ,
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