A note on embedding into product spaces

Using factorization properties of an operator ideal over a Banach space, it is shown how to embed a locally convex space from the corresponding Grothendieck space ideal into a suitable power of E, thus achieving a unified treatment of several embedding theorems involving certain classes of locally convex spaces.     

Some new results on ε-Tensor products of locally convex spaces

The paper contains the proofs of three new propositions on ε-Tensor products of locally convex spaces. The first two of these propositions are on the ε-tensor product of inductive limits of locally convex spaces. The third proposition is on integral bilinear forms. For inductive tensor products and π-tensor products, some results on properties of permanence appear in A. Grothendieck’s famous thesis. We prove, in the present paper, some properties of permanence of ε-tensor products.     

The injective hull op an operator ideal on locally convex spaces

In his book (3), Pietsch presents the problem of a direct construction of the injective hull of an operator ideal on loca. lly convex spaces. We construct the infective hull of an arbitrary operator ideal on locally convex spaces using the functor, where and B cover the family of all equicontinuous subsets of E'(L is the category of all locally convex spaces). If the operator ideal is bounded, we ob tain its infective hull using seminorm ideals.     

核局部凸空间的一个新特征

Ａ．Ｐｉｅｔｓｃｈ＾［１］在讨论核局部凸空间时给出了两类矢值序列空间ｌ１［Ｘ］和ｌ１｛Ｘ｝。本文建立了矢值序列空间ｌ１［Ｘ］及ｌ１｛Ｘ｝和连续线性算子空间Ｌ（ｃ０，Ｘ）及绝对可和算子空间ＡＳ（ｃ０，Ｘ）之间的拓扑同胚关系。通过ｃ０上的矢值算子类Ｌ（ｃ０，Ｘ）和ＡＳ（ｃ０，Ｘ）及其上的拓扑等价关系，对局部凸空间Ｘ是核空间给出了一个新的特征刻划。     

Nuclear cones in product spaces,pareto efficiency and Ekeland-type variational principles in locally convex spaces

In this article, we present a method to obtain new Ekeland-type variational principles for mappings defined between locally convex spaces. The method is based on a special nuclear cone defined in a product of two locally convex spaces and on Pareto efficiency with respect to this cone.     

A property of Dunford-Pettis type in topological groups

The property of Dunford-Pettis for a locally convex space was introduced by Grothendieck in 1953. Since then it has been intensively studied, with especial emphasis in the framework of Banach space theory.

In this paper we define the Bohr sequential continuity property (BSCP) for a topological Abelian group. This notion could be the analogue to the Dunford-Pettis property in the context of groups. We have picked this name because the Bohr topology of the group and of the dual group plays an important role in the definition. We relate the BSCP with the Schur property, which also admits a natural formulation for Abelian topological groups, and we prove that they are equivalent within the class of separable metrizable locally quasi-convex groups.

For Banach spaces (or for metrizable locally convex spaces), considered in their additive structure, we show that the BSCP lies between the Schur and the Dunford-Pettis properties.

     

Factoring operators over Hilbert-Schmidt maps and vector measures

We study the structure of Banach spaces X determined by the coincidence of nuclear maps on X with certain operator ideals involving absolutely summing maps and their relatives. With the emphasis mainly on Hilbert-space valued mappings, it is shown that the class of Hilbert—Schmidt spaces arises as a ‘solution set’ of the equation involving nuclear maps and the ideal of operators factoring through Hilbert—Schmidt maps. Among other results of this type, it is also shown that Hilbert spaces can be characterised by the equality of this latter ideal with the ideal of 2-summing maps. We shall also make use of this occasion to give an alternative proof of a famous theorem of Grothendieck using some well-known results from vector measure theory.     

New existence results for efficient points in locally convex spaces ordered by supernormal cones

In this paper, the definition of supernormality for convex cones in locally convex spaces is discussed in detail on many interesting examples. Starting from the new direction for the study of the existence of efficient points (Pareto type optimums) in locally convex spaces offered by the concept of supernormal (nuclear) cone, we establish some existence results for the efficient points using boundedness and completeness of conical sections induced by non-empty subsets and we specify properties for the sets of efficient points beside important remarks     

Eine Charakterisierung der Boole'schen Algebra zu einer co -direkten Zerlegung

In this paper nuclear Boolean Algebras of projections in a locally convex space are considered. This are Boolean Algebras with special continuity properties, which are shared, for instance, by each bounded Boolean Algebra of projections in an ?^∞-space and by the algebra of each equicontinuos spectral measure in a nuclear space. It will be shown that a ?-complete nuclear Boolean Algebra leads to a c_o-direct sum of locally convex spaces and all the projections of the algebra belong to the complete algebra of projections of this c_o-direct partition. On the other hand if in a given locally convex space E there exists a nuclear complete Boolean Algebra of projections which has multiplicity one then each equicontinuos Boolean Algebra of projections in E is nuclear.     

Grothendieck space ideals and weak continuity of polynomials on locally convex spaces

We introduce the classes of locally convex spaces with the local Dunford-Pettis property and locally dual Schur spaces. We examine their properties and their relationship to other classes of locally convex spaces. In the class of locally convex spaces with the local Dunford-Pettis property all polynomials are weakly sequentially continuous whereas in the class of locally dual Schur spaces all polynomials are weakly continuous on bounded sets.     

Convergent star products for projective limits of Hilbert spaces

Given a locally convex vector space with a topology induced by Hilbert seminorms and a continuous bilinear form on it we construct a topology on its symmetric algebra such that the usual star product of exponential type becomes continuous. Many properties of the resulting locally convex algebra are explained. We compare this approach to various other discussions of convergent star products in finite and infinite dimensions. We pay special attention to the case of a Hilbert space and to nuclear spaces.     

The Levy-Steinitz rearrangement theorem for duals of metrizable spaces

Extending the Levy-Steinitz rearrangement theorem in ℝ ⁿ , which in turn extended Riemann’s theorem, Banaszczyk proved in 1990/93 that a metrizable, locally convex space is nuclear if and only if the domain of sums of every convergent series (i.e. the set of all elements in the space which are sums of a convergent rearrangement of the series) is a translate of a closed subspace of a special form. In this paper we present an apparently complete analysis of the domains of sums of convergent series in duals of metrizable spaces or, more generally, in (DF)-spaces in the sense of Grothendieck. The research of the first author was partially supported by DGES Project PB97-0333, and the second one by a grant of the Ministerio de Educatión y Cultura of Spain (Ref: Sab 1995-0736).     

Grothendieck space ideals and weak continuity of polynomials on locally convex spaces

We introduce the classes of locally convex spaces with the local Dunford-Pettis property and locally dual Schur spaces. We examine their properties and their relationship to other classes of locally convex spaces. In the class of locally convex spaces with the local Dunford-Pettis property all polynomials are weakly sequentially continuous whereas in the class of locally dual Schur spaces all polynomials are weakly continuous on bounded sets. Research supported by Science Foundation Ireland, Basic Research Grant 2004.     

Convolution operators on spaces of entire functions

We show that nontrivial convolution operators on certain spaces of entire functions on E are frequently hypercyclic when E is a normed space and when E is the strong dual of a Fréchet nuclear space. We also obtain results of existence and approximation for convolution equations on certain spaces of entire functions on arbitrary locally convex spaces.     

Convolution on distribution spaces characterized by regularization

Locally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space of test functions into a given locally convex lattice of measures. The convolutor spaces are endowed with the topology of uniform convergence on bounded sets. Their locally convex structure is characterized via regularization and function-valued seminorms under mild structural assumptions on the space of measures. Many recent generalizations of classical distribution spaces turn out to be special cases of the general convolutor spaces introduced here. Recent topological characterizations of convolutor spaces via regularization are extended and improved. A valuable property of the convolutor spaces in applications is that convolution of distributions inherits continuity properties from those of bilinear convolution mappings between the locally convex lattices of measures.     

一类局部凸空间及其映照特征

称局部凸空间（E,(?)0）为WCM空间若对于任何弱于(?)0的局部凸拓扑(?),（E,(?)）与（E,(?)0）具相同的弱紧圆凸集.本文研究了WCM空间的存在性及其与其他类型局部凸空间之间的关系,还给出了WCM空间的一种映照特征.     

A Frobenius Theorem for Locally Convex Global Analysis

 We prove a Frobenius theorem for Banach space complemented subbundles of the tangent bundle of a manifold modelled on locally convex spaces. The proof is based on an implicit function theorem for maps from locally convex spaces to Banach spaces proved in a recent paper of the author.     

An Implicit Function Theorem in Locally Convex Spaces

The notion of strict differentiability in BANACH spaces is generalized to locally convex spaces in such a way, that computations in the bornological operator ideal of bounded operators play the role of norm estimations. We get a remarkably rich theory including not only the easy formulas like the chain rule, but also the deep theorems like the implicit function theorem. The BANACH space proofs can be translated almost literally.     

Characterizations of ultrabarrelledness and barrelledness involving the singularities of families of convex mappings

Summary The paper reveals that ultrabarrelled spaces (respectively barrelled spaces) can be characterized by means of the density of the so-called weak singularities of families consisting of continuous convex mappings that are defined on an open absolutely convex set and take values in a locally full ordered topological linear space (respectively locally full ordered locally convex space). The idea to establish such characterizations arose from the observation that, in virtue of well-known results, the density of the singularities of families of continuous linear mappings allows to characterize both the ultrabarrelled spaces and the barrelled spaces.     

Diestel-Faires定理在局部凸空间中的推广

通过Banach 空间与局部凸空间的对比,将Banach 空间上的Diestel-Faires 定理在局部凸空间上进行推广。进一步给出了局部凸空间上的Orlicz-Pettis定理与推论。