Carácter topológico de Isomorfismo del Borel

Let E be a DFN space, E' its strong topological dual, H(E) and H(E') the corresponding spaces of holomorphic functions from E and E' into endowed with the respective compact-open topologies and Exp E' the vector subspace of H(E') consisting of the entire functions of exponential type on E'. In this article we endow Exp E' with a vector space topology and show that Borel´s Transformation is an isomorphism between the topological vector spaces h(E') (strong topological dual of H(E)) and Exp E'

The symmetric tensor product of a direct sum of locally convex spaces

An explicit representation of the n-fold symmetric tensor product (equipped with a natural topology tau such as the projective, injective or inductive one) of the finite direct sum of locally convex spaces is presented. The formula for circle times(tau,delta)(n)(F-1 circle plus F-2) gives a direct proof of a recent result of Diaz and Dineen land generalizes it to other topologies tau) that the n-fold projective symmetric and the n-fold projective "full" tensor product of a Iocally convex space fare isomorphic if E is isomorphic to its square E-2

Tensor topologies on spaces of symmetric tensor products

Here we show how, in the general context of locally convex spaces, it is possible to get an n-tensor topology (on spaces of n-tensor products) from an n-tensor topology on spaces of symmetric n-tensors products. Indeed, given an n-tensor topology on the spaces of symmetric n-tensor products we construct an n-tensor topology on the spaces of all n-tensor products whose restriction to the symmetric ones gives the original topology. Moreover, we prove that when one starts with an n-tensor topology, restricts it to symmetric tensors and then extends it, the original topology is obtained when it is symmetric, and we also obtain some results on complementation with applications to spaces of polynomials. Part of these results generalize to the context of locally convex spaces some Floret's results in [17] and [18]

Una nota sobre las topologías To y Tw en H(U)

Sean, U un abierto equilibrado de un espacio localmente convexo complejo E, H(U) el espacio vectorial de las funciones holomorfas en U y to y tw las topologías compacto-abierta y portada de Nachbin respectivamente en H(U). Se prueba en esta nota que si E es de Fréchet-Montel, entonces to = tw en H(U) si to y tw coinciden en todos los espacios de polinomios homogéneos continuos en E. Además se relaciona el problema de conocer si to=tw en esos espacios con un clásico problema de Grothendieck relativo al e-producto de dos espacios de tipo DFM

Locally determining sequences in infinite-dimensional spaces.

A subset L of a complex locally convex space E is said to be locally determining at 0 for holomorphic functions if for every connected open 0-neighborhood U and every f∈H(U), whenever f vanishes on U∩L, then f≡0. The authors' main result is that if E is separable and metrizable, then every set which is locally determining at 0 contains a null sequence which is also locally determining at 0. This answers a question of J. Chmielowski [Studia Math. 57 (1976), no. 2, 141–146;], who was the first to study locally determining sets. The proof of the main theorem makes use of the following result of K. F. Ng [Math. Scand. 29 (1971), 279–280;]: Let E be a normed space with closed unit ball BE. Suppose that there is a Hausdorff locally convex topology τ on E such that (BE,τ) is compact. Then E with its original norm is the dual of the normed space F={φ∈E∗: φ|BE is τ-continuous}, with norm ∥φ∥=sup{|φ(x)|: x∈BE

(BB) properties on Fréchet spaces

In this paper we point out some relations concerning properties (BB)(n) and (BB)(n,s) related to the "Probleme des topologies" of Grothendieck and give a first example of a Frechet space with the (BB)(2) property but without the (BB)(3) property

On the "three-space problem" for spaces of polynomials.

A property P of locally convex spaces is called a three-space property whenever the following implication holds: if both a closed subspace F and the corresponding quotient E/F of a locally convex space E have P then E has P as well. The authors consider properties P of the form: E has P whenever two "natural'' topologies coincide on the spaces of n-homogeneous polynomials on E. They consider topologies of the uniform convergence on all absolutely convex compact or bounded subsets as well as the strong topology and the Nachbin ported topology. The results obtained are mostly negative and the counterexamples are variations of the known spaces

Propiedades (BB)n y topologías en P(nE)

Properties (BB)n, n = 2, 3, ... on a locally convex space (see the definition below) have been recently introduced ([10]). They are interesting, among other things, in connection with the study of natural topologies on spaces of polynomials, multilinear and holomorphic mappings. As it is proved in [1] there are Fr´echet spaces with the (BB)2 property but without the (BB)3 property. Here, for a given n = 3, ... we get an space without the property (BB)n+1 and study an equivalent condition for that space to have the (BB)n propert

Relations between T0 and Tω on spaces of holomorphic functions

In this paper we give a survey on recent results concerning the coincidence or not of the compact-open and the Nachbin ported topologies on the space of all holomorphic functions on an open subset of a complex Fréchet space.

On the quasinormability of Hb(U).

Let E be a complex locally convex space, U a balanced open subset of E. The author continues the study of the quasinormability of the space Hb(U) of all holomorphic functions on U which are bounded on the U-bounded subsets of U. His main goal is to prove the following theorem: "For all Fréchet spaces E with the (BB)∞-property, Hb(U) is quasinormable for every balanced open subset U of E''; this includes all the known results about the quasinormability of Hb(U) in the Fréchet space settin