Mário C. Matos

On Holomorphy Versus Linearity in Classifying Locally Convex Spaces

Publisher Summary This chapter provides an overview of the fact that in the holomorphic approach the corresponding concepts have been introduced as holomorphically bornological, holomorphically barreled, holomorphically infrabaralled, and holomorphically Mackey spaces that are more restricted classes than the corresponding linear ones. In the linear theory of locally convex spaces, it is classical to study bornological, barreled, infrabarreled, and Mackey spaces. An interesting highlight is the holomorphic Banach-Steinhaus on a Frechet space that contains, as a particular case, the classical linear Banach-Steinhaus theorem on such a space. A holomorphically bornological space is also a bornological space. A semimetrizable space is a holomorphically bornological space. A Silva space is known to be essentially the same thing as the dual of a Frechet-Schwartz space, or FS-space; thus, it is also known as a DFS-space. A Silva space is a holomorphically bornological space. Any inductive limit of bornological spaces is a bornological space.

Lineability of summing sets of homogeneous polynomials

Given a continuous n-homogeneous polynomial P : E ⟹ F between Banach spaces and 1 ≤ q ≤ p < ∞, in this article we investigate some properties concerning lineability and spaceability of the (p; q)-summing set of P, defined by S p; q (P) = {a ∈ E : P is (p; q)-summing at a}.

Convolution equations in spaces of infinite dimensional entire functions

Abstract We prove general results of surjectivity for convolution equations in spaces of entire functions in locally convex spaces. These results improve and partially unify known results due to Berner, Boland, Dwyer, Gupta and Matos. We also obtain results on approximation of solutions.

Reinhardt domains of holomorphy in Banach spaces

Abstract It is known that in c 0 the holomorphic functions representable by multiple power series (the series of monomials) are those of nuclear type. In this article we show that in a Banach space with unconditional Schauder basis every domain of existence is the domain of existence of a holomorphic function representable by a multiple power series and, in fact, the domain of convergence of such a series.

On Convolution Operators in Spaces of Entire Functions of a Given Type and Order

Publisher Summary This chapter discusses the convolution operators in spaces of entire functions of a given type and order. It introduces the spaces of entire functions of order k ϵ [ 1, + ∞] and type strictly less than A ϵ (0,+ ∞) in normed spaces. The corresponding spaces for which the type is allowed to be equal to A were introduced for k ϵ [ 1,+ ∞ ] and A ϵ [ 0, + ∞ ). All these spaces carry natural locally convex topologies and they are the infinite dimensional analogues of the spaces considered by Martineau. This chapter studied the Fourier-Borel transformations in these spaces and is able to identify algebraicaly and topologically the strong duals of these spaces with other spaces of the same kind.

On holomorphy and compactness in Banach spaces

I f E and F are complex Banach spaces and U is a non-void open subset of E, 9((U; F) denotes the complex vector space of all holomorphic mappings from U into F. There are several locally convex topologies which may be considered on 9((U; F). Among them some are natural in the sense tha t they coincide with the usual topology on 9g(U; F) for E finite dimensional. This paper presents results about the characterization of the relatively compact subsets of 9(~(U; F) for one of these topologies. Theorems of Nachbin and Aron appear as special eases of these results. Let ~ be the family of all continuous mappings ~ from U into R such t h a t O < ~(x) ~ d ( x , a U ) = d i s t a n c e of x to the boundary aU of U. For each ~E let ~ ( U ) be the collection of all the finite unions of closed balls /~Q(x) with center x E U and radius ~ < ~(x). 9d~(U; F) denotes the complex vector space of all the holomorphic mappings from U into F which are bounded over the elements of c~(U). In c)~( U;/v) we consider the topology of uniform convergence over the elements of ~ ( U ) .

Convolution Equations in Infinite Dimenstions: Brief Survey, New Results and Proofs

Abstract In the last fifteen years a large amount of results were obtained on convolution equations in normed and locally convex spaces. The aim of this work is to contribute to the improvement and clarification of the theory by presenting new results and connections between previously known theorems. For convenience and necessity of presentation we recall most existing results and give their references, so that this paper is also a brief survey on the subject.

On Separately Holomorphic and Silva Holomorphic Mappings

Publisher Summary This chapter presents a discussion on separately holomorphic and Silva holomorphic mappings. E, F, and G denote complex locally convex spaces. The subsets U ⊂ E and V ⊂ F indicate non-void open subsets. β E denote the family of all bounded closed absolutely convex subsets o f E. If B ∊ β E let E B denote the vector subspace of E generated by B and normed by the Minkowsky functional ║ . ║ B associated to B. It is recalled that a subset K of E is a strict compact set it there is B in β B such that K is contained and compact in E B .