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Proc. Amer. Math. Soc. 43 (1974), 337–340. P. Lotz, Uniform convergence of operators on L∞ and similar spaces. Math. Z. 190 (1985), 207–220. A. Albanese, J. J. G. Meise, D. Vogt, Introduction to Functional Analysis. Clarendon Press, Oxford, 1997. [18] G. B. Moscatelli, On the space p+ = q>p q . Math. Nachr. 147 (1990), 7–12. [19] S. Okada, Spectrum of scalar–type spectral operators and Schauder decompositions. Math. Nachr. 139 (1988), 167–174. [20] P. P´erez Carreras, J. Bonet, Barrelled Locally Convex Spaces.

5] there is a positive Radon measure ν on V such that ϕ(s) = ρ(s)dν(ρ). V For every γ ∈ Γ we obtain n n cj ck ϕγ (sj s∗k ) = V t∈S j,k=1 cj ρ(sj )|2 ≥ 0. aγ,t ρ(t)| j=1 Because for every continuous function f : V → C such that f ≥ 0 the function can be uniformly approximated on V by functions of the type √ f n cj ρ(sj ) j=1 this means that we have V t∈S aγ,t ρ(t)f (ρ)dν(ρ) ≥ 0 for every continuous function f : V → C such that f ≥ 0. Consequently for every γ ∈ Γ the measure aγ,t ρ(t)) · ν (ρ → t∈S is positive which means that ν{ρ ∈ V| aγ,t ρ(t) < 0} = 0.

Now let (x, y) ∈ G(A). Then there exist xn ∈ D such that xn → x and Axn → y in X. It follows that xn = A−1 (Axn ) → A−1 y in D. Hence x = A−1 y ∈ D and Ax = y. 1. 1. (ii)⇒(i). Assume condition (ii). Then A is invertible. It follows that the graph norm x A := Ax (x ∈ D) defines an equivalent norm on D. In fact, since A is an isometric isomorphism from (D, · A ) to X, it follows that (D, · A ) is complete. Since x A ≤ A x D , it follows from the Open Mapping Theorem that both norms on D are equivalent.

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A Baire category approach in existence theory of differential equations by Pianigiani G.


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