Vector-valued meromorphic functions

Abstract

A locally complete locally convex space E satisfies that every weakly meromorphic function defined on an open subset of C with values in E is meromorphic if and only if E does not contain a countable product of copies of C. A characterization of locally complete spaces in the spirit of known characterizations of the (metric) convex compactness property is also given.The purpose of this article is to show that every weakly meromorphic function defined on an open subset Ω of C with values in a locally complete locally convex space E is meromorphic if and only if every very strongly convergence sequence in E in the sense of Dineen [6, Definition 3.23] has only finitely many elements different from 0. According to [19] this condition is equivalent to the non-containment of a countable product of copies of C. A Fréchet space satisfies this condition if and only if it has a continuous norm, and every (DF)-space has this property too. The representation of the space of meromorphic functions with values in E as an ε-product of Schwartz is analyzed. The following result which could be of independent interest is also obtained: a locally convex space E is locally complete if and only if the closed absolutely convex hull of the image of a compact subset of R n by a weakly C 1 function with values in E is compact. The corresponding statement for continuous functions on compact sets instead of weakly C 1 functions characterizes a strictly stronger completeness condition [20].Our notation for locally convex spaces and functional analysis is standard. We refer to [15,17,18]. We recall the terminology which will be repeatedly used. In a metric space we denote by B(a, r), D(a, r) and S(a, r) the open ball, the closed ball and the sphere centered on a with radius r respectively. Throughout this paper E denotes a complex locally convex space. Let I be an index set, the product of locally convex spaces each one of them isomorphic to E is denoted by E I , and their direct sum is denoted by E (I ) . C N is denoted by ω and C (N) by ϕ.The problem of finding conditions in locally convex spaces to ensure that weak holomorphy implies holomorphy goes back to Dunford (cf. [7]). Recent results obtained in this direction can be found in [1,2,8,10]. The aim of this paper is to clarify the relation between (vectorvalued) meromorphic functions and weakly meromorphic functions.

Weighted spaces of vector-valued functions and the $$\varepsilon$$-product

“…Indeed, the first definition implies the second since every convex hull of a set A ⊂ E is contained in its absolutely convex hull. On the other hand, we have acx(A) = cx(ch(A)) by [ [7,35]. In addition, we remark that every semi-Montel space is semireflexive by [17, 11.5 Since weighted spaces of continuously partially differentiable vector-valued functions will serve as our standard examples, we recall the definition of the spaces C k (Ω, E) .…”

“…Thus Theorem 14 (iii) settles the case for k < ∞ . If k = ∞ and E is locally complete, we observe that K ∶= acx(( ) E f (K)) for f ∈ CW ∞ (Ω, E) is absolutely convex and compact by [7,Proposition 2,p. 354].…”

Extension of vector-valued holomorphic and harmonic functions

“…[12,28]). Moreover, to illustrate the scope of Theorem 9, we mention that it gives a direct proof of the fact that weak-C ∞ implies C ∞ : Let f : Ω → E be a map into a locally complete space E such that u • f ∈ C ∞ (Ω) for all u ∈ E ′ .…”

Topologies on spaces of vector-valued meromorphic functions

“…The space of E-valued holomorphic functions on ft is denoted by //(ft, E). We refer to [6,Theoreme 2], [1,Section 3], [11, 7.4] and [5, 4.3] for equivalent definitions of vector-valued holomorphic and meromorphic functions (see also [3,Chapter II,2]). …”

“…In [2], Bonet, Maestre and the author proved that if £ is a locally complete locally convex space which does not contain a; as a subspace then M{£1, E) can be canonically identified with the e-product of Schwartz M(£2)eE. We consider in M(Q, E) the topology which makes this representation a topological isomorphism.…”