Vol. 77, No. 1, 1978

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On the strong compact-ported topology for spaces of holomorphic mappings

M. Bianchini, O. W. Paques and M. C. Zaine

Vol. 77 (1978), No. 1, 33–49

Suppose E is a separated complex locally convex space, U is non void open subset of E, F a complex normed space and (U;F) the complex vector space of all holomorphic mappings from U into F. On (U;F) we consider the following topologies; a) τωs, the topology generated by the seminorms p which are K B ported for some K U compact and B E bounded. A seminorm p is K B ported if for every 𝜖 > 0, with K + 𝜖B U, there is c(𝜖) > 0, such that p(f) c(𝜖)sup{∥f(x);x K + 𝜖B} for all f ∈ℋ(U;F); b) τ0, the compact open topology; c) τs the topology defined by J. A. Barroso in “Topologias nos espaços de aplicações holomorfas entre espaços localmente convexos”, An. Acad. Brasil. Ci, 43, 1971. The topology τωs is an generalization of the Nachbin topology (L. Nachbin, Topology on Spaces of Holomorphic Mappings, Springer-Verlag, 1968). The following results are valid: 1. ℋ⊂ℋ(U;F) is τ0-bounded if, and only if, is τωs-bounded. 2. ℋ⊂ℋ(U;F) is τωs-relatively compact if, and only if, is τs-relatively compact. 3. Let E be a quasi complete space. Then τ0 = τωs on (E;C) if, and only if E is a semi-Montel space. Moreover, the completion of (E;C) on the τωs topology and the bornological topology associated to τ0 are characterized via the Silva-holomorphic mappings.

Mathematical Subject Classification 2000
Primary: 46G20
Secondary: 32A30, 46E10
Received: 29 November 1976
Revised: 14 October 1977
Published: 1 July 1978
M. Bianchini
O. W. Paques
M. C. Zaine