Vollständige und B-vollständige topologische Vektorgruppen Graphensätze und Open-Mapping Theorems

We are studying complete and B-complete topological vector groups. These Objects have been introduced by P. Kenderov [6] and D. A. Raikov [11]. They form a category TVG intermediate to the categories of topological Abelian groups and topological vector spaces and are close enough to the last one to give many useful applications to it. We first consider the problem of completion in the most used subcategories of TVG. A special functor allows to play back permanence property questions of completeness in locally convex vector groups to the same questions for locally convex vector spaces. Some examples of complete locally convex vector groups follow. We then unify some differently defined notions of B-completeness and generalize well known theorems concerning B-complete locally convex topological vector spaces to locally convex topological vector groups. Barrelledness concepts introduced in 9 and a special functor constructed in section 6 are used to formulate analogues of the closed graph and open mapping theorem for locally convex vector groups. The remainder of the note is left for applications to locally convex vector spaces. Many theorems about 1^p-sums of normed spaces are proved, as well as the B-completeness of a vast class of locally convex vector spaces including the spaces and of Köthe ([7], §13, No 5,6).     

Higher derivatives of mappings of locally convex spaces

We establish sufficient conditions for n-fold bounded differentiability (b-differentiability) of mappings of locally convex spaces and sufficient conditions for n-fold Hyers-Lang differentiability (HL-differentiability) of mappings of pseudotopological linear spaces. We describe a class of locally convex spaces on which there exist everywhere infinitely b-differentiable real functions which are not everywhere continuous (and so are not everywhere HL-differentiable). Our results show, in particular, that for a wide class of locally convex spaces a significant number of the known definitions of C-mappings fall into two classes of equivalent definitions.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 729–744 November, 1977.     

<Emphasis Type="Italic">C</Emphasis>-distribution semigroups and <Emphasis Type="Italic">C</Emphasis>-ultradistribution semigroups in locally convex spaces

The main purpose of this paper is to study C-distribution semigroups and C-ultradistribution semigroups in the setting of sequentially complete locally convex spaces. We provide a few important theoretical novelties in this field and some interesting examples. Under consideration are stationary dense operators in a sequentially complete locally convex space.     

The Lipschitzianity of convex vector and set-valued functions

It is well known that every scalar convex function is locally Lipschitz on the interior of its domain in finite dimensional spaces. The aim of this paper is to extend this result for both vector functions and set-valued mappings acting between infinite dimensional spaces with an order generated by a proper convex cone C. Under the additional assumption that the ordering cone C is normal, we prove that a locally C-bounded C-convex vector function is Lipschitz on the interior of its domain by two different ways. Moreover, we derive necessary conditions for Pareto minimal points of vector-valued optimization problems where the objective function is C-convex and C-bounded. Corresponding results are derived for set-valued optimization problems.     

Regular Maximal Monotone Operators

The purpose of this paper is to introduce a class of maximal monotone operators on Banach spaces that contains all maximal monotone operators on reflexive spaces, all subdifferential operators of proper, lsc, convex functions, and, more generally, all maximal monotone operators that verify the simplest possible sum theorem. Dually strongly maximal monotone operators are also contained in this class. We shall prove that if T is an operator in this class, then (the norm closure of its domain) is convex, the interior of co(dom(T)) (the convex hull of the domain of T) is exactly the set of all points of at which T is locally bounded, and T is maximal monotone locally, as well as other results.     

On some relationships between biorthogonal systems,linear topologies and the weak‐AFPP

In this paper, we obtain new results for the weak‐AFPP in abstract spaces by exploiting biorthogonal systems techniques. Firstly, we investigate the strong‐AFPP on countably infinite dimensional Hausdorff locally convex spaces. Spaces of this class are shown to be sequentially complete iff they have the hereditary FPP for totally bounded, closed convex sets. This might open a research line for the analysis of weak‐AFPP in such frames. In connection, we provide a simple criterion for the containement of ?₁‐sequences in terms of strongly‐equicontinuous biorthogonal systems. We then establish a few results concerning the existence of Hausdorff finer vector topologies on abstract spaces having as prescribed condition the existence of such systems. The proofs are based on methods of Peck and Porta concerning building of finer vector topologies, and a classical construction of Singer which allows us to prove under rather natural conditions the existence of equicontinuous biorthogonal systems in metrizable locally convex spaces. These results are compatible with the failure of the weak‐AFPP. We also study the inverse problem by proving that every infinite dimensional vector space admits a (non‐locally convex) Hausdorff vector topology which is complete, non‐metrizable and is compatible with a bounded Hamel Schauder basis. It is shown further that such a topology has the ‐AFPP, where is the linear span of coefficient functionals associated to a Hamel basis. Finally, inspired by a result of Shapiro, we observe that if X is a non‐locally convex F‐space with an absolute basis, then the weak‐AFPP is equivalent to the fact that every bounded convex subset of X is compact.     

Invariance of the order and type of a sequence of operators

In the paper, the invariance property of characteristics (the order and type) of an operator and of a sequence of operators with respect to a topological isomorphism is proved. These characteristics give precise upper and lower bounds for the expressions ‖A_n(x)‖_p and enable one to state and solve problems of operator theory in locally convex spaces in a general setting. Examples of such problems are given by the completeness problem for the set of values of a vector function in a locally convex space, the structure problem for a subspace invariant with respect to an operator A, the problem of applicability of an operator series to a locally convex space, the theory of holomorphic operator-valued functions, the theory of operator and differential-operator equations in nonnormed spaces, and so on. However, the immediate evaluation of characteristics of operators (and of sequences of operators) directly by definition is practically unrealizable in spaces with more complicated structure than that of countably normed spaces, due to the absence of an explicit form of seminorms or to their complicated structure. The approach that we use enables us to find characteristics of operators and sequences of operators using the passage to the dual space, by-passing the definition, and makes it possible to obtain bounds for the expressions ‖A_n(x)‖_p even if an explicit form of seminorms is unknown.     

Über assoziierte lineare und lokalkonvexe Topologien

Certain properties E of linear topological or locally convex spaces induce a functor in the corresponding category, which assigns to every space (X,F) an associated topologyF ^E. The well-known notions of the coarsest barrelled topology stronger than a given locally convex topology or of the strongest locally convex topology weaker than a given linear topology are examples of this concept. In the first two parts of this paper we consider the problem, whether the above functors commute with other processes, such as forming products, linear and locally convex direct sums, inductive limits and completions. With help of two technical lemmas we prove in the third part, that every separated locally convex space is a quotient of a complete locally convex space, in which every bounded set has a finite dimensional linear span. This sharpens results of Y. Kōmura [12], M. Valdivia [18] and W.J. Wilbur [20].     

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.     

The structure of functions satisfying the law of large numbers in a class of locally convex spaces

For each function that satisfies the law of large numbers with values in a certain class of locally convex spaces with the Radon-Nikodym property the following decomposition holds: , where is integrable by seminorm, and is a Pettis integrable function which is scalarly 0.

     

Permanence properties of C(X,E)

Let C(X,E) be the vector space of all continuous functions on a completely regular Hausdorff space X with values in a locally convex space E, equipped with the compact-open topology. In this note it will be shown that for many classes of locally convex spaces K C(X,E) lies in K if and only if C(X)=C(X,¦K) (¦K=¦R or C) and E belong to K. This is valid for the classes K. of all metrizable, normed, (DF)-, -locally topological, separable, quasi-complete, complete, nuclear, Schwartz, semi-Montel, Montel, semi-reflexive, reflexive and quasi-normable locally convex spaces, respectively. But in general C(X,E) is not quasi-barrelled, barrelled and bornological, respectively, if C(X) and E belong to the same class. We shall give sufficient conditions for C(X,E) to be quasi-barrelled and barrelled, respectively.Herrn Professor Gottfried Köthe zum 75. Geburtstag gewidmet     

Closed graph and open mapping theorems for topological $ \tilde \lc $‐modules and applications

We present closed graph and open mapping theorems for ‐linear maps acting between suitable classes of topological and locally convex topological ‐modules. This is done by adaptation of De Wilde's theory of webbed spaces and Adasch–Ernst–Keim's theory of barrelled spaces to the context of locally convex and topological ‐modules, respectively. We give applications of the previous theorems to Colombeau theory as well to the theory of Banach ‐modules. In particular we obtain a necessary condition for ??^∞‐hypoellipticity on the symbol of a partial differential operator with generalized constant coefficients (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximation properties in tensor products

For distinct classes of locally convex spaces and tensor topologies α =? and α = π it is proved that \(E\hat \otimes _\alpha F\) has the approximation property if and only if E and F have this property.     

Lokalkonvexe Unterräume in Topologischen Vektorräumen und Das ε-Produkt

It is well-known that the algebraic tensor product E Y of a not necessarily locally convex topological vector space E and a locally convex space Y can be identified with a subspace of the so-called -product EY (a space of continuous linear mappings from Y into E). So, whenever EY is complete, even the completed tensor product is (isomorphic to) a subspace of EY. As this occurs in many important cases, it is interesting to remark that, for each continuous linear operator u from a locally convex space F into E, there exists a locally convex U with continuous embedding jUE and a continuous linear map ûFU such that u=j·û. As main applications of a combination of these ideas, we obtain a characterization of the functions in as continuous functions with values in locally convex spaces (this gives new aspects for the intergration theory of Gramsch [5]) and a result extending a theorem in [6] on holomorphic functions with values in non locally convex spaces to arbitrary complex manifolds.     

On metrizability of compactoid sets in non-archimedean locally convex spaces

In 2003, N. De Grande-De Kimpe, J. Kąkol and C. Perez-Garcia using t-frames and some machinery concerning tensor products proved that compactoid sets in non-archimedean (LM)-spaces (i.e. the inductive limits of a sequence of non-archimedean metrizable locally convex spaces) are metrizable. In this paper we show a similar result for a large class of non-archimedean locally convex space with a £-base, i.e. a decreasing base (U_α)_αN^N of neighbourhoods of zero. This extends the first mentioned result since every non-archimedean (LM)-space has a £-base. We also prove that compactoid sets in non-archimedean (DF)-spaces are metrizable.     

The semi-<Emphasis Type="Italic">E</Emphasis> cone convex set-valued map and its applications

In this paper, firstly, a new notion of the semi-E cone convex set-valued map is introduced in locally convex spaces. Secondly, without any convexity assumption, we investigate the existence conditions of the weakly efficient element of the set-valued optimization problem. Finally, under the assumption of the semi-E cone convexity of set-valued maps, we obtain that the local weakly efficient element of the set-valued optimization problem is the weakly efficient element. We also give some examples to illustrate our results.     

Semistable laws on topological vector spaces

Summary In this paper we discuss three types of results: Firstly, we present two Lévy-Hinin type representations of Poisson type infinitely divisible (i.d.) laws on certain topological vector (TV) spaces; one of these complements a known representation due to Dettweiler. Secondly, we define and characterize r-semistable laws on locally convex TV spaces and also obtain good representation of their characteristic functions. Finally, we discuss the existence and the continuity of the semigroup { _tt>0} of i.d. laws on locally convex TV spaces. These complement similar known results of Siebert.The research of this author is partially supported by the Office of Naval Research under contract No. N0014-78-C-0468     

Exponentiation in V-categories

For a Heyting algebra V which, as a category, is monoidal closed, we obtain characterizations of exponentiable objects and morphisms in the category of V-categories and apply them to some well-known examples. In the case these characterizations of exponentiable morphisms and objects in the categories (P)Met of (pre)metric spaces and non-expansive maps show in particular that exponentiable metric spaces are exactly the almost convex metric spaces, while exponentiable complete metric spaces are the complete totally convex ones.     

A Hille-Yosida theorem for Bi-continuous semigroups

In order to treat one-parameter semigroups of linear operators on Banach spaces which are not strongly continuous, we introduce the concept of bi-continuous semigroups defined on Banach spaces with an additional locally convex topology . On such spaces we define bi-continuous semigroups as semigroups consisting of bounded linear operators which are locally bi-equicontinuous for and such that the orbit maps are -continuous. We then apply the result to semigroups induced by flows on a metric space as studied by J. R. Dorroh and J. W. Neuberger.     

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.