Suppose E is a separated
complex locally convex space, U is non void open subset of E, F a complex
normed space and ℋ(U;F) the complex vector space of all holomorphic
mappings from U into F. On ℋ(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) τ∞s 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, ℋ is τωs-bounded. 2. ℋ⊂ℋ(U;F) is τωs-relatively
compact if, and only if, ℋ is τ∞s-relatively compact. 3. Let E be a quasi
complete space. Then τ0= τωs on ℋ(E;C) if, and only if E is a semi-Montel
space. Moreover, the completion of ℋ(E;C) on the τωs topology and the
bornological topology associated to τ0 are characterized via the Silva-holomorphic
mappings.
