6 MATI ABEL AND REYNA MAR´ IA PEREZ-TISCARE˜´ NO Following, we prove that (K(R,A [−i, i]) × A, τi ) is complete. For this, for every fixed i ∈ N let ((fk,ak)) be a Cauchy sequence in (K(R,A [−i, i]) × A, τi ). Then, for each fixed n ∈ N and arbitrary ε 0, there exists N 0 such that Qi,n((fk,ak) − (fl,al)) ε whenever k l N. Hence, qi,n(fk − fl) ε and pn(ak − al) ε whenever k l N. So (fk) and (ak) are Cauchy sequences in the complete algebras K(R,A [−i, i]) and A respectively. Let f ∈ K(R,A [−i, i]) and a ∈ A be such that limk→∞ fk = f and limk→∞ ak = a. Then, ((fk,ak)) converges to (f, a). Indeed, for each fixed n ∈ N and arbitrary ε 0 there exists N such that qi,n(fk − f) ε 2 and pn(ak − a) ε 2 whenever k N. Thus Qi,n((fk,ak) − (f, a)) ε for each n ∈ N whenever k N. Consequently, (K(R,A [−i, i]) × A, τi ) is complete and hence a locally pseudoconvex F -algebra. Therefore K(R,A)×A in the locally pseudoconvex inductive limit topology τ is a LFp-algebra5. To show that K(R,A) × A is a Q-LFp-algebra, let τ be the topology on K(R,A) × A given by kn-homogeneous seminorms Qn, where Qn(ψ, a) = qn(ψ) + pn(a) and qn(ψ) = sup(pn(ψ(x))) x∈R for every n ∈ N, ψ ∈ K(R,A) and a ∈ A. Then (K(R,A) × A, τ ) is a metrizable locally pseudoconvex algebra. We prove that the inclusion Ψi : (K(R,A [−i, i]) × A, τi ) → (K(R,A) × A, τ ) is continuous in this topology for each i. For this, let Urn = {ψ ∈ K(R,A) : qn(ψ) r} and Wrn = {a ∈ A : pn(a) r} for each r 0 and n ∈ N. Then Vrn = Urn × Wrn is a neighborhood of (θ, θA) (θ is the zero element of K(R,A) and θA the zero element of A) in (K(R,A)×A, τ ) and Vrn ∩ (K(R,A [−i, i]) × A) = {ψ ∈ K(R,A [−i, i]) : qi,n(ψ) r} × Wrn is a neighborhood of (θ, θA) in K(R,A [−i, i]) × A in the topology τi (θ is the zero element also in K(R,A [−i, i])). Hence Ψi is continuous at (θ, θA) for every i. Therefore Ψi is continuous. So τ ≺ τ by the definition of τ . Thus, it is enough to show that (K(R,A) × A, τ ) and (K(R,A [−i, i]) × A, τi ) are Q-algebras for every i ∈ I. By assumption, (A, τ) is a Q-algebra. Then there are n ∈ N and ε 0 such that eA + Wεn ⊂ Inv(A) and Vεn = Uεn × (eA + Wεn) is a neighborhood of (θ, eA) in (K(R,A) × A, τ ), hence in (K(R,A) × A, τ ) also because τ ≺ τ . To see that V ε 2 n ⊂ Inv(K(R,A) × A), let (ψ, a) ∈ V ε 2 n. Then pn(ψ(x) + a − eA) ≤ pn(ψ(x)) + pn(a − eA) ε 2 + ε 2 = ε 5We consider on K(R, A)×A and on K(R, A [−i, i])×A the separately continuous multiplica- tion defined by (f, a)(f , a ) = (ff +fa +af , aa ), where fa (x) = f(x)a and af (x) = a(f (x)). In that case (θ, eA) is the unit element in K(R, A) × A and K(R, A [−i, i]) × A, where θ is the zero element in K(R, A [−i, i]) and eA is the unit element in A.

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