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 uν ∈ Eα for every ν with 1 ν n. Hence x ∈ Eα 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 Oα be for every α ∈ I an absolutely kα-convex neighborhood of zero in Eα such that Oα ⊆ 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 Eβ by hypothesis and fβα0 (Oα0 ) ⊆ fβ −1 [fβ ◦ fβα0 (Oα0 )] = fβ −1 [fα0 (Oα0 )] ⊆ fβ −1 (O) for every β ∈ I. To show that O ⊂ QinvE, let x ∈ O. Then x = n ν=1 μν xν 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 xν = fα0 (oν) for each ν. Hence x = fα0 n ν=1 μν oν ⊆ 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 = −→ Eα to Eα. Corollary 2.5. Let (E, τ) be a LFpg-algebra defined by locally pseudoconvex F -algebras Eα with α ∈ I and let ρEα be a seminorm on Eα 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 ··· ,

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2016 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.