2 MATI ABEL AND REYNA MAR´ IA PEREZ-TISCARE˜´ NO U = Γk(U) for some k ∈ (0, 1]. When kλ = k for each λ ∈ Λ (that is, k does not depend on Uλ) or inf{kλ : λ ∈ Λ} = k 0, then E is a locally k-convex algebra. The topology of a locally pseudoconvex algebra E can be defined by a family P = {pλ : λ ∈ Λ} of kλ-homogeneous seminorms (that is, pλ(μa) = | μ |kλ pλ(a) for λ ∈ Λ and a ∈ E), where kλ ∈ (0, 1] for each λ ∈ Λ, and pλ(a) = inf{| μ |kλ : a ∈ μΓkλ (Uλ)} for each a ∈ A and λ ∈ Λ (see, for example, [17, pp. 3–6] or [8, pp. 189 and 195]). 2. We recall first what means an inductive limit of algebras. For it, let I be a (non-empty) directed set with the order “ ≺ . So, for any α, β ∈ I there is a γ ∈ I such that α ≺ γ and β ≺ γ. Let (Eα)α∈I be a family of algebras and for every α, β ∈ I with α ≺ β let {fβα : Eα → Eβ} be a family of homomorphisms such that the following properties have been satis- fied: 1) fαα = idEα for every α ∈ I. 2) fγα = fγβ ◦ fβα for any α, β, γ ∈ I such that α ≺ β ≺ γ. In case, when every Eα is a unital algebra with unit element eα, then it is assumed that fβα(eα) = eβ whenever α ≺ β. The family of algebras (Eα)α∈I with maps fβα, defined above, is an inductive (or directed) system of algebras Eα and it is denoted by (Eα,fβα). Let E0 = ∑ α Eα (a disjoint union). Elements x, y ∈ E0 (then x ∈ Eα and y ∈ Eβ for some α, β ∈ I) are called equivalent (in short x ∼ y), if there exists γ ∈ I such that α ≺ γ, β ≺ γ and fγα(x) = fγβ(y). The quotient set (E0/∼) is the inductive (or direct) limit of the inductive system (Eα,fβα). We shall denote this, as usual, by lim(Eα,fβα) −→ or simply by limEα. −→ For every α ∈ I, let iα : Eα → E0 be the inclusion and π : E0 → E0/∼ the quotient map. Then, fα = π ◦ iα : Eα → E = limEα −→ for every α ∈ I is a homomorphism (later on, it will be called a canonical map from Eα to E). When the inductive limit limEα −→ has the unit element e, then it is assumed that fα(eα) = e for each α ∈ I. It is known that (1) E = α∈I fα(Eα) (see [15, p. 110]). Moreover, fβ ◦ fβα = fα whenever α ≺ β and fα(Eα) ⊆ fβ(Eβ) for any α, β ∈ I with α ≺ β. Therefore, the algebraic operations can be given on limEα −→ (see [15, p. 111]). If inductive limits of topological algebras (Eα,τα) are considered, it is assumed that the homomorphisms fβα : Eα → Eβ (α, β ∈ I with α ≺ β) are continuous

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