Schauder decompositions and functional calculus

We establish a functional calculus, with nice properties, for one and several continuous operators on some non-normable locally convex spaces, more specifically, for operators on the space of entire functions and on other power series spaces. In particular we obtain spectral mapping theorems. The calculus rests on Schauder decompositions for the spaces under consideration, which are of independent interest.     

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.     

DIFFERENTIABILITY OF CONVEX FUNCTIONS ON SUBLINEAR TOPOLOGICAL SPACES AND VARIATIONAL PRINCIPLES IN LOCALLY CONVEX SPACES

This paper presents a type of variational principles for real valued w lower semicon-tinuous functions on certain subsets in duals of locally convex spaces, and resolve a problem concerning differentiability of convex functions on general Banach spaces. They are done through discussing differentiability of convex functions on nonlinear topological spaces and convexification of nonconvex functions on topological linear spaces.     

Superefficiency in Local Convex Spaces

In the framework of normed spaces, Borwein and Zhuang introduced superefficiency and gave its concise dual form when the underlying decision problem is convex. In this paper, we consider four different generalizations of the Borwein and Zhuang superefficiency in locally convex spaces and give their concise dual forms for convex vector optimization. When the ordering cone has a base, we clarify the relationship between Henig efficiency and the various kinds of superefficiency. Finally, we show that whether the four kinds of superefficiency are equivalent to each other depends on the normability of the underlying locally convex spaces.     

Phelps' lemma, Danes' drop theorem and Ekeland's principle in locally convex spaces

A generalization of Phelps' lemma to locally convex spaces is proven, applying its well-known Banach space version. We show the equivalence of this theorem, Ekeland's principle and Danes' drop theorem in locally convex spaces to their Banach space counterparts and to a Pareto efficiency theorem due to Isac. This solves a problem, concerning the drop theorem, proposed by G. Isac in 1997.

We show that a different formulation of Ekeland's principle in locally convex spaces, using a family of topology generating seminorms as perturbation functions rather than a single (in general discontinuous) Minkowski functional, turns out to be equivalent to the original version.

     

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.     

Some results on non-semireflexive spaces

In this paper we prove a result similar to a generalization due to M. Valdivia (4) of a theorem of V.L. Klee (1), in the context of real locally convex spaces which possess a metrizable locally convex topology coarser than the Mackey topology. In particular, we obtain some criteria of reflexivity for metrizable locally convex spaces. A particular case works for complex spaces.We thank Prof. Valdivia for his help and constant orientation This work follows from his suggestions.     

Hypercyclic operators on quasi-Mazur spaces

Modifying the method of Ansari, we give some criteria for hypercyclicity of quasi-Mazur spaces. They can be applied to judging hypercyclicity of non-complete and non-metrizable locally convex spaces. For some special locally convex spaces, for example, Köthe (LF)-sequence spaces and countable inductive limits of quasi-Mazur spaces, we investigate their hypercyclicity. As we see, bounded biorthogonal systems play an important role in the construction of Ansari. Moreover, we obtain characteristic conditions respectively for locally convex spaces having bounded sequences with dense linear spans and for locally convex spaces having bounded absorbing sets, which are useful in judging the existence of bounded biorthogonal systems.     

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.     

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 STRONG OPTIMIZATION THEOREM IN LOCALLY CONVEX SPACES

This paper presents a geometric characterization of convex sets in locally convex spaces on which a strong optimization theorem of the Stegall-type holds, and gives Collier's theorem of w Asplund spaces a localized setting.     

The law of large numbers in locally convex spaces

Summary The law of large numbers is extended to random elements taking values in locally convex spaces. The necessary and sufficient conditions for the law are given in a large class of locally convex spaces, vix. normed spaces. This class includes, among others, the test function spaces and the distribution spaces.     

On Solidness of Polar Cones

We investigate the properties of cones whose polars are solid in different polar topologies. By a standard duality argument, we obtain a number of necessary and sufficient conditions for closed convex cones to be solid in various locally convex spaces. From this, we can deduce easily the extensions of previous related results. Furthermore, we construct a class of closed convex cones in some Banach spaces, which are not solid but whose polars satisfy the angle property. This solves the Han conjecture in the negative.     

Multidirectional Mean Value Inequalities in Quasidifferential Calculus

In the current paper, a Clarke–Ledyaev type mean value inequality is proved for semicontinuous functions defined in a Banach space that are quasidifferentiable in the sense of Demyanov–Rubinov. A stronger variant valid under compactness assumption in separable spaces and extensions for functions with semicontinuous Dini derivatives in locally uniformly convex Banach spaces and with merely bounded Dini derivatives are then established. Subsequently, applications of these mean value inequalities to solvability of nonsmooth parametric equations and to the estimation of local and global Hoffman error bound for inequalities are investigated via a decrease principle.     

Fredholm theory forp-adic locally convex spaces

Summary The authors develop the Fredholm theory for semi-compact operators on non-archimedean locally convex spaces. This theory coincides with Schikhof's Fredholm theory for compact operators on Banach spaces which fails for non-complete normed 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.     

Uniquely generated Grothendieck space ideals

Starting from a result of Pietsch on the Grothendieck ideal of strictly nuclear locally convex spaces, we classify all those Grothendieck ideals of nuclear locally convex spaces that are generated by a unique operator ideal on the class of Banach 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。     

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。     

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.