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

1978; Mathematical Sciences Publishers; Volume: 77; Issue: 1 Linguagem: Inglês

10.2140/pjm.1978.77.33

1945-5844

Mauro Bianchini, Otília W. Paques, Maria Carmelina F. Zaine,

Homotopy and Cohomology in Algebraic Topology

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) τ ωs , the topology generated by the seminorms p which are K -B ported for some KczU compact and BcE bounded.A seminorm p is K -B ported if for every ε >0, with K + εB c U 9 there is c(ε) > 0, such that P(f)^c(ε) sup{\\f(x)\\;xeK-fεB} for all fe^(U F); b) r 0 , the compact open topology; c) τ TOS the topology defined by J. A. Barroso in "Topologias nόs espaςos de aplicaςδes holomorfas entre espaςos localmente convexos", An.Acad.Brasil.Ci, 43, 1971.The topology τ ωs is an generalization of the Nachbin topology (L.Nachbin, Topology on Spaces of Holomorphic Mappings, Springer-Verlag, 1968).The following results are valid: 1. ^c £t?{JJ\ F) is τ 0 -bounded if, and only if, <%? is τ^-bounded.2. ^a^f{TJ\F) is τ^-relatively compact if, and only if, £? is r^-relatively compact.3. Let E be a quasi complete space.Then τ 0 = τ m on <%f(E m ,C) if, and only if E is a semi-Montel space.Moreover, the completion of &?(E\ C) on the τ m topology and the bornological topology associated to τ 0 are caracterized via the Silvaholomorphic mappings.