An Abstract Banach-Steinhaus Theorem and Applications to Function Spaces

We give an abstract Banach-Steinhaus theorem for locally convex spaces having suitable algebras of linear projections modelled on a σ-finite measure space. This theorem is applied to deduce barrelledness results for the space L∞ (μ, E) of essentially bounded and μ-measurable functions from a Radon measure space (Ω, σ, μ) into a locally convex space E and also for B (μ, E), the closure of the space of simple functions. Sample: if μ is atomless, then B (μ, E) is barrelled if and only if E is quasi-barrelled and E′(β (E′, E)) has the property (B) of Pietsch.     

Certain classes of continuous linear operations

Certain classes of continuous linear operators in Banach and locally convex spaces are studied. A characterization of operators T: X Y, transforming bounded sets of the Banach space X into conditionally weakly compact sets of the Banach space Y, is given, and also a particular case where X = C(K) is considered. It is proved that if E is a Fréchet space and F is a complete ()-space, then the classes of absolutely summing and Nikodýmizing operators from E into F coincide.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 285–296, February, 1978.     

Barrelledness and bornological conditions on spaces of vector-valued μ-simple functions

Let S(μ, E) be the space of (classes of μ-a.e. equal) simple functions defined on a (non-trivial) measure space with values in a locally convex space E. The following results hold: S(μ,E) is quasi-barrelled (resp. bornological) if and only if E is quasi-barrelled (resp. bornological) and E′(β(E′,E)) has the property (B) of Pietsch; S(μ, E) is barrelled if and only if S(μ,K) is barrelled and E is barrelled and nuclear; S(μ, E) is never ultrabornological; and S(μ, E) is a DF-space if and only if E is a DF-space.     

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].     

Banach Disks and Marrelledness Properties of Metrizalbe <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript> (Ω, <Emphasis Type="Italic">X</Emphasis>)

Let be a non-empty set and X a metrizable locally convex space. We show that the metrizable locally convex space c₀ (, X) is p-barrelled (totally barrelled) if and only if X is p-barrelled (totally barrelled). Some applications for closed graph theorems are included.     

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.     

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.     

Der Graphensatz von A. P. und W. Robertson für s-Rädme

A locally convex space E is said to be an s-space [1] if every closed linear map of E onto a barrelled space is open. The aim of this paper is to replace the B-complete spaces in the closed graph theorem of A. P. and W. ROBERTSON [7] by s-spaces. Previous work of PTAK [5] and PERSSON [4] implies that B-complete spaces and t-polar spaces are s-spaces. Thus our result includes that of ADASCH [2] who generalized Robertson's theorem, taking t-polar spaces instead of B-complete ones.     

A class of bornological barrelled spaces which are not ultrabornological

N. Bourbaki, [1, p. 35], notices that it is not known if every bornological barrelled space is ultrabornological. In this paper we prove that if E is the topological product of an infinite family of bornological barrelled spaces, of non-zero dimension, there exists an infinite number of bornological barrelled subspaces ofE, which are not ultrabornological. We also give some examples of barrelled normable spaces which are not ultrabornological.Supported in part by the Patronato para el Fomento de la Investigación en la Universidad.     

Über die Tonneliertheit von lokalkonvexen Tensorprodukten

It is known that the inductive tensor product of two barrelled spaces is barrelled and that the projective tensor product of two barrelled metrizable spaces or barrelled (DF)-spaces is barrelled. In this note it will be shown by a counterexample that the projective tensor product EF of two barrelled spaces E and F in general is not barrelled, even if E is (DF)-Montel-space and F (F)-Montel-space. Furthermore we show that the -tensor product of two (B)-spaces in general is not barrelled. It follows from the fact that an (F)-space E is nuclear if and only if the -tensor product E l ⁴ is barrelled.     

C ϰ(X) and a property of (db)-spaces

Summary A new class of locally convex linear topological spaces, the (db)-spaees, recently defined by Robertson, Tweddle and Yeomans, interpolates the classes of Baire-like and unordered Baire-like spaces. Saxon and Narayanaswami have given convex metrizable spaces that distinguish among these classes. This paper gives a new characterization of (db)-spaces from which is extracted the class of b-spaces. This class interpolates the classes of -spaces and -spaces of Lehner. Let C(X) be the space of all real-valued continuous functions on the completely regular Sausdorf space X, supplied with the topology of uniform convergence on compact sets of X. It is shown that C(X) is a b-space if and only if it is an -space. A characterization of X for which C(X) is a (db)-space is unknown. Other open questions are stated in the paper.     

Barrelled spaces with a B-complete completion

Let (E, ) be a barrelled locally convex space. The aim of this paper is to describe the barrelled topologies on E weaker than . When the completion Ê of (E, ) is B-complete, various properties of the barrelled topologies weaker than are proved. Some examples are given to illustrate the possible situations.     

The integral representation of vector measures on a completely regular space

We consider the vector space C(X, E) of all bounded continuous functions from a completely regular space X into a Banach space E. It is given a special locally convex topology . We prove the analog of the Riesz-Markov theorem for the -continuous linear operators which map C (X, E) into a Banach space F.Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 401–408, September, 1976.     

Positive approximate identities and lattice-ordered dual spaces

It is shown that if E[7] is a locally convex space ordered by a closed, generating positive cone K satisfying certain mild hypotheses relating K and 7, then E′ is lattice-ordered by the dual cone K′ whenever the identity linear transformation on E is the pointwise limit of sums of transformations x→〈x,x′〉z where z∈K and x′∈K′. The converse is true for certain classes of spaces, e.g., Fréchet spaces.     

A space of vector-valued measures and a strict topology

Let X be a completely regular Hausdorff space and let E be a real locally convex Hausdorff space. Katsaras [2] has studied the topologies ₀, , and ₁, for the vector-valued case on C_rc(X,E), the space of all continuous E-valued functions on X with relatively compact range. The corresponding dual spaces are the spaces M_t (B,E'), M (B,E'), and M (B,E') of all t-additive, all -additive, and all -additive members of M(B,E'), the dual space of C_rc (X,E') under the uniform topology. In this paper we study the subspace M_e(B,E') of M(B,E'). A locally convex topology _e is defined on C_rc(X,E) that yields M_e (B,E') as a dual space. It is proved that if E is strongly Mackey then (C (X,E),_e) is strongly Mackey.The author is grateful to Professor J. Schmets for useful suggestions.     

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.     

TOPOLOGICAL DECOMPOSITIONS OF THE DUALS OF LOCALLY CONVEX VECTOR SEQUENCE SPACES

ABSTRACT

Let Λ be a scalar sequence space which is endowed with a normal locally convex topology. For a separated locally convex space E we denote by Λ(E) the vector space of all sequences g in E for which (>g(i),a<) ε Λ for all a ε E'. We define a locally convex topology ζ on Λ(E) and then characterize the dual of the ζ-closure (denoted by Λ_c (E)) of the finite sequences in Λ(E). We demonstrate the existence of a continuous projection from Λ(E)' onto a subspace of Λ(E)' which is isomorphic to Λ_c(E)'. Furthermore, we find a topological decomposition of Λ^α _c (E)”, where one of the factors is isomorphic to Λ;^α(E). These results are then applied to find necessary and sufficient conditions for Λ^α(E) to be semi-reflexive. A parallel development yields the same results for the space Λ(E') of all sequences f in E' for which (>x, f(i)<) ε Λ; for all x ε E, when E is barrelled. We conclude the paper by application of the results on vector sequence spaces to spaces of operators—including for instance, necessary and sufficient conditions for L_b (E,Λ;) and L_b (Λ,E) to be semi-reflexive.     

Asplund operators and holomorphic maps

LetE andF be locally convex topological vector spaces. A holomorphic mapf: E→F is defined to be an Asplund map if it takes the separable subsets of a neighbourhood of eacha∈E into absolutely convex weakly metrisable subsets ofF; a Banach space is an Asplund space if and only if its identity map has this property. We show that a continuous linear map from a quasinormable locally convex spaceE into a Banach spaceF is an Asplund map if and only if it factors through an Asplund space. IfE andF are both Banach spaces, then a holomorphic mapf: E→F is an Asplund map if and only if its derivative maps factor through Asplund spaces for eacha∈E. This is true if and only if such a factorisation holds ata=0. Part of this research was done during a visit to the University of Namibia, whose financial support is gratefully acknowledged This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990     

Very Convex Banach Spaces

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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.