Frechet-Montel Spaces Research Articles

An Inverse Mapping Theorem in Fréchet-Montel Spaces

Influenced by a recent note by M. Ivanov and N. Zlateva, we prove a statement in the style of Nash-Moser-Ekeland theorem for mappings from a Frechet-Montel space with values in any Frechet space (not necessarily standard). The mapping under consideration is supposed to be continuous and directionally differentiable (in particular Gateaux differentiable) with the derivative having a right inverse. We also consider an approximation by a graphical derivative and by a linear operator in the spirit of Graves’ theorem. Finally, we derive corollaries of the abstract results in finite dimensions. We obtain, in particular, sufficient conditions for the directional semiregularity of a mapping defined on a (locally) convex compact set in directions from a locally conic set; and also conditions guaranteeing that the nonlinear image of a convex set contains a prescribed ordered interval.

Topological properties of strict (LF)-spaces and strong duals of Montel strict (LF)-spaces

Following Banakh and Gabriyelyan (Monatshefte Math 180:39–64, 2016), a Tychonoff space X is Ascoli if every compact subset of $$C_k(X)$$ is equicontinuous. By the classical Ascoli theorem every k-space is Ascoli. We show that a strict (LF)-space E is Ascoli iff E is a Frechet space or $$E=\phi $$ . We prove that the strong dual $$E'_\beta $$ of a Montel strict (LF)-space E is an Ascoli space iff one of the following assertions holds: (i) E is a Frechet–Montel space, so $$E'_\beta $$ is a sequential non-Frechet–Urysohn space, or (ii) $$E=\phi $$ . Consequently, the space $$\mathcal {D}(\varOmega )$$ of test functions and the space of distributions $$\mathcal {D}'(\varOmega )$$ are not Ascoli that strengthens results of Shirai (Proc Jpn Acad 35:31–36, 1959) and Dudley (Proc Am Math Soc 27:531–534, 1971), respectively.

On a problem of topologies in infinite dimensional holomorphy

The authors solve an interesting open problem concerning the equivalence of the compact-open topology τ0 and the Nachbin ported topology τω on spaces of holomorphic functions. (See, for example, the book by S. Dineen [Complex analysis in locally convex spaces, North-Holland, Amsterdam, 1981; MR0640093 (84b:46050)] for background.) Let H(U) denote the space of complex-valued holomorphic functions on an open subset U of a complex Frechet-Montel space F. Ansemil and S. Ponte [Arch. Math. (Basel) 51 (1988), no. 1, 65–70; MR0954070 (90a:46109)] showed that these two topologies agree on H(U) for balanced U if and only if, for every natural number n, P(nF) is a Montel space. Using this result, they showed that for balanced open subsets U of certain non-Schwartz, Frechet-Montel spaces, τ0=τω. Earlier, J. Mujica [J. Funct. Anal. 57 (1984), no. 1, 31–48; MR0744918 (86c:46050)] had shown that τ0=τω for Frechet-Schwartz spaces. It is not hard to see that the two topologies differ if F is not Montel. The authors' counterexample is the Frechet-Montel space F of Taskinen [Studia Math. 91 (1988), no. 1, 17–30; MR0957282 (89k:46087)]. The authors observe that the complete symmetric projective tensor product Fs⊗ˆπF contains an isomorphic copy of l1. Consequently, P(2F) cannot be Montel, and the result follows.

The compact open and the Nachbin ported topologies on spaces of holomorphic functions

Some examples of Frechet Montel spaces E, which are not Schwartz spaces, for which the compact open and the Nachbin ported topologies coincide on the space of all holomorphic functions on an arbitrary balanced open subset of E, are given.

Generating Frechet-Montel spaces that are not Schwartz by closed linear operators

Generating Frechet-Montel spaces that are not Schwartz by closed linear operators

The Approximation Property and Locally Convex Spaces Defined by the Ideal of Approximable Operators

The connection between the approximation property and certain classes of locally convex spaces associated with the ideal 𝔊 of approximable operators will be discussed. It will be shown that a Frechet Montel space has the approximation property iff it is a mixed 𝔊-space, or equivalently, iff its strong dual is a 𝔊-space. Similarly, a Silva space has the approximation property iff it is a mixed 𝔊-space or equivalently iff it is a 𝔊-space. We shall also show that a Frechet Schwartz space with the bounded approximation property is always a 𝔊-space.

Holomorphic functions on strict inductive limits

We study strict inductive limits of Frechet Montel (FM) spaces and reflexive Frechet (RF) spaces and we obtain some interesting examples in the theory of infinite dimensional holomorphy.