A free convenient vector space for holomorphic spaces

Abstract

Abstract.In this paper we show that there exists a free convenient vector space for the case of holomorphic spaces and holomorphic maps. This means that for every space X with a holomorphic structure, there exists an appropriately complete locally convex vector space 2X and a holomorphic maplx:X ~.X, such that for any vector space of the same kind the map (tx)* : L(2X, E) ~ ~(X, E) is a bijection. Analogously to the smooth case treated in [-2, 5.1.1] the free convenient vector space 2X can be obtained as the Mackey closure of the linear subspace spanned by the image of the canonical map X ~ ~(X, C)'.In the second part of the paper we prove that in the case where X is a Riemann surface, one has 2X = ~(X, C)'.

The convenient setting for non-quasianalytic Denjoy–Carleman differentiable mappings

“…We expect λ M (R) to be equal to C M (R, R) ′ as it is the case for the analogous situation of smooth mappings, see [19, 23.11], and of holomorphic mappings, see [25] and [26]. Proof.…”

“…That also (4) describes the same bornology follows again by the S-uniform boundedness principle, since the inductive limit in (4) is regular by what we said before for the special case E = R and hence the structure of (4) is convenient. We expect λ M (R) to be equal to C M (R, R) as it is the case for the analogous situation of smooth mappings, see [19, 23.11], and of holomorphic mappings, see [25] and [26].…”

Bornologicity of certain spaces of bounded linear operators

“…In the context of holomorphic spaces, we have a very similar situation, as shown in [13], where the analogon of Proposition 2.2 is established for Riemann surfaces.…”

The convenient setting for quasianalytic Denjoy–Carleman differentiable mappings

“…We expect λ M (R) to be equal to C M (R, R) ′ as it is the case for the analogous situation of smooth mappings, see [15, 23.11], and of holomorphic mappings, see [23] and [24]. Proof.…”