This article is devoted to a study of locally convex topologies on $\mathbf{H}\left(U\right)$ (where $U$ is an open subset of the locally convex topological vector space $E$ and $\mathbf{H}\left(U\right)$ is the set of all complex valued holomorphic functions on $E$). We discuss the following topologies on $\mathbf{H}\left(U\right)$:(a) the compact open topology ${\mathbf{I}}_0$,(b) the bornological topology associated with ${\mathbf{I}}_0$,(c) the ported topology of Nachbin ${\mathbf{I}}_{\omega }$,(d) the bornological topology associated with ${\mathbf{I}}_{\omega }$ ; and(e) the ${\mathbf{I}}_{\omega }$ topological of Nachbin.For $U$ balanced we show these topologies are...

In this article we discuss the relationship between domains of existence domains of holomorphy, holomorphically convex domains, pseudo convex domains, in the context of locally convex topological vector spaces. By using the method of Hirschowitz for ${\Pi }_{n=1}^{\infty }\mathbf{C}$ and the method used for Banach spaces with a basis we prove generalisations of the Cartan-Thullen-Oka-Norguet-Bremmerman theorem for finitely polynomially convex domains in a variety of locally convex spaces which include the following:1) $N$-projective...

In this article we show that a number of apparently different properties coincide on the set of holomorphic functions on a strict inductive limit (all inductive limits are assumed to be countable and proper) of Banach spaces and that they are all satisfied only in the trivial case of a strict inductive limit of finite dimensional spaces. Thus the linear properties of a strict inductive limit of Banach spaces rarely translate themselves into holomorphic properties.

Let $E$ and $F$ be two complex Banach spaces, $U$ a nonempty subset of $E$ and $K$ a compact subset of $E$. The concept of holomorphy type $\theta$ between $E$ and $F$, and the natural locally convex topology ${𝒯}_{\omega ,\theta }$ on the vector space ${\mathscr{H}}_{\theta }(U,F)$ of all holomorphic mappings of a given holomorphy type $\theta$ from $U$ to $F$ were considered first by L. Nachbin. Motived by his work, we introduce the locally convex space ${\mathscr{H}}_{\theta }(K,F)$ of all germs of holomorphic mappings into $F$ around $K$ of a given holomorphy type $\theta$, and study its interplay with ${\mathscr{H}}_{\theta }(U,F)$ and some...

The classical Erdös spaces are obtained as the subspaces of real separable Hilbert space consisting of the points with all coordinates rational or all coordinates irrational, respectively. One can create variations by specifying in which set each coordinate is allowed to vary. We investigate the homogeneity of the resulting subspaces. Our two main results are: in case all coordinates are allowed to vary in the same set the subspace need not be homogeneous, and by specifying different sets for different...

Let $𝔐$ be a Jordan-Banach algebra with identity 1, whose norm satisfies:(i) $\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel$,   $a,b\in 𝔐$(ii) $\parallel a^2\parallel =\parallel a{\parallel }^2$(iii) $\parallel a^2\parallel \le \parallel a^2+b^2\parallel$.$𝔐$ is called a JB algebra (E.M. Alfsen, F.W. Shultz and E. Stormer, Oslo preprint (1976)). The set ${𝔐}^{+}$ of squares in $𝔐$ is a closed convex cone. $(𝔐,{𝔐}^{+},\mathbf{1})$ is a complete ordered vector space with $\mathbf{1}$ as a order unit. In addition, we assume $𝔐$ to be monotone complete (i.e. $𝔐$ coincides with the bidual ${𝔐}^{**}$), and that there exists a finite normal faithful trace $\phi$ on $𝔐$.Then the completion $\{{𝔐}^{+}{\}}_{\phi }$ of ${𝔐}^{+}$ with respect to the Hilbert structure...

We give a survey of our recent results on homological properties of Köthe algebras, with an emphasis on biprojectivity, biflatness, and homological dimension. Some new results on the approximate contractibility of Köthe algebras are also presented.

Extending previous results of H. Salas we obtain a characterisation of hypercyclic weighted shifts on an arbitrary F-sequence space in which the canonical unit vectors $\left(e_n\right)$ form a Schauder basis. If the basis is unconditional we give a characterisation of those hypercyclic weighted shifts that are even chaotic.

Se introducen los ideales de operadores que admiten extensión o levantamiento y se presenta una nueva aproximación al estudio de la escisión de sucesiones exactas cortas de espacios de Banach. Se considera la maximalidad de estos ideales y se investiga si son cerrados respecto de los límites puntuales acotados. Se resumen algunos ejemplos y se clarifica el papel de los espacios L1 y L∞.