Positive linear extension operators for spaces of affine functions

The following result is proved: letE be anF-space (that is, the space of all continuous affine functions defined on a compact universal cap van shing at zero) and letMχE be anM-ideal. Then, ifE/M is a π₁ with positive defining projections, then there is a positive linear operator ϱ:E/M→E of norm one such that ϱ lifts the canonical mapE→E/M. In the proof, which heavily depends on work of Ando, we study ensor products of certain convex cones with compact bases, and we calculate the norm of a positive linear operator defined on a finite dimensional space with range in aF-space. Various corollaries are deduced for split faces of compact convex sets and for morphisms ofC ^*-algebras.     

DIEUDONNÉ—KÖTHE DUALITY FOR VECTOR—VALUED FUNCTION SPACES: LOCALIZATION OF BOUNDED SETS AND BARRELLEDNESS

Abstract

We study Dieudonné-Köthe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.     

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.     

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.     

Intersections of Fréchet Schwartz spaces and their duals

In this article the structure of the intersections of a Fréchet Schwartz space F and a (DFS)-space E=ind _n E _n is investigated. A complete characterization of the locally convex properties of E ⋃ F is given. This space is boraological if and only if the inductive limit E + F is complete. The results are based on recent progress on the structure of (LF)-spaces. The article includes examples of (FS)-spaces F and (DFS)-spaces E such that there are sequentially continuous linear forms on E ⋃ F which are not continuous, thus answering a question of Langenbruch. Acknowledgement: The results in this article were obtained during the author’s stay at the University of Paderborn, Germany, during the academic year 1994/95. The support of the Alexander von Humboldt Stiftung is greatly appreciated. The content of the article was presented as an invited paper in a Special Session of the AMS meeting in New York in April, 1996.     

Spaces of Vector-Valued Integrable Functions and Localization of Bounded Subsets

We study the structure of bounded sets in the space L¹{E} of absolutely integrable Lusin-measurable functions with values in a locally convex space E. The main idea is to extend the notion of property (B) of Pietsch, defined within the context of vector-valued sequences, to spaces of vector-valued functions. We prove that this extension, that at first sight looks more restrictive, coincides with the original property (B) for quasicomplete spaces. Then we show that when dealing with a locally convex space, property (B) provides the link to prove the equivalence between Radon–Nikodym property (the existence of a density function for certain vector measures) and the integral representation of continuous linear operators T: L¹ → E, a fact well-known for Banach spaces. We also study the relationship between Radon–Nikodym property and the characterization of the dual of L¹{E} as the space L^∞{E′_b}.     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous. (Received 24 March 1999; in final form 14 February 2000)     

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     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous.     

The Hahn-Banach Property and the Axiom of Choice

We work in set theory ZF without axiom of choice. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gateaux-differentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F → ? is a linear functional such that f ≤ p|F then there exists a linear functional g : E → ? such that g extends f and g ≤ p. We also prove that the continuous Hahn-Banach property on a topological vector space E is equivalent to the classical geometrical forms of the Hahn-Banach theorem on E. We then prove that the axiom of Dependent choices DC is equivalent to Ekeland's variational principle, and that it implies the continuous Hahn-Banach property on Gateaux-differentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous Hahn-Banach property, they do not satisfy the whole Hahn-Banach property in ZF+DC.     

Holomorphic functional calculus for operators on a locally convex space

One-variable holomorphic functional calculus is studied on the bornological algebra Lec(E) of all continuous linear oprators on a complete locally convex space E. It is proven that the following three basic notions of the theory are equivalent: (i) existence of projective resolvent of an operator T at a point λ₀, (ii) strict regularity of λ₀ for the operator T in the sense of [12, 13, 15], (iii) tamability of the operator (λ₀ ? T)^?1 (T if λ₀ = ∞), which means that there is a new equivalent system of seminorms on E, such that the operator is bounded in each of them.     

Uniform approximation: The non-locally convex case

IfS is a compact Hausdorff space of finite covering dimension and (E, τ) is a real or complex topological vector space (not necessarily locally convex), we prove a Weierstrass-Stone theorem for subsets ofC(S;E), the space of all continuous functions fromS intoE, equipped with the topology of uniform convergence overS.     

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].     

Sufficient conditions for existence of extrema in linear programming

It is shown that, on a closed convex subset X of a real Hausdorff locally convex space E, a continuous linear functional x′ on E has an extremum at an extreme point of X, provided X contains no line and X ∩ (x′)^?1 (λ₀) is non-empty and weakly compact for some real λ₀. It is also shown that any weakly locally compact closed convex subset of E that contains no line is the sum of its asymptotic cone and the closed convex hull of its extreme points.     

Boundedly Compatible Webs and Strict Mackey Convergence

We look for characterizations of those locally convex spaces that satisfy the strict Mackey convergence condition within the context of spaces with webs. We will say that a locally convex space has a boundedly compatible web if it has a web of absolutely convex sets whose members behave like zero neighborhoods in a metrizable locally convex space. It will be shown that these locally convex spaces satisfy the strict Mackey convergence condition. One consequence of this result will be a characterization of boundedly retractive inductive limits. We will also prove that if E is locally complete and webbed, then the strict Mackey convergence condition is equivalent to E having a boundedly compatible web.     

Countable-Codimensional Subspaces of c0-Barrelled Spaces

A Hausdorff locally convex space is said to be c₀-barrelled (respectively ω-barrelled) if each sequence in the dual space that converges weakly to 0 (respectively that is weakly bounded), is equicontinuous. It is proved that if a c₀-barrelled space E has dual E′ weakly sequentially complete, then every subspace of countable codimension of E is c₀-barrelled. We prove that the hypothesis on E′ cannot be dropped and we supply an example of a complete c₀-barrelled space with dual weakly sequentially complete that is not ω-barrelled.     

The Choquet integral representation in the affine vector-valued case

LetX be any compact convex subset of a locally convex Hausdorff space andE be a complex Banach space. We denote byA(X, E) the space of all continuous and affineE-valued functions defined onX. In this paper we prove thatX is a Choquet simplex if and only if the dual ofA(X, E) is isometrically isomorphic by a selection map toM _m (X, E*), the space ofE*-valued,w*-regular boundary measures onX. This extends and strengthens a result of G. M. Ustinov. To do this we show that for any compact convex setX, each element of the dual ofA(X, E) can be represented by a measure inM _m (X, E*) with the same norm, and this representation is unique if and only ifX is a Choquet simplex. We also prove that ifX is metrizable andE is separable then there exists a selection map from the unit ball of the dual ofA(X, E) into the unit ball ofM _m (X, E*) which is weak* to weak*-Borel measurable.This work will constitute a portion of the author's Ph.D. Thesis at the University of Illinois.     

A Remark on the Local Lipschitz Continuity of Vector Hysteresis Operators

It is known that the vector stop operator with a convex closed characteristic Z of class C ¹ is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping n is Lipschitz continuous on the boundary Z of Z. We prove that in the regular case, this condition is also necessary.     

Convex-compact sets and Banach discs

Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual E′ of a locally convex space E is the σ(E′,E)-closure of the union of countably many σ(E′,E)-relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.     

On linear selections of convex set-valued maps

We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x ₁ + x ₂) ⊂ φ(x ₁) + φ(x ₂). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: X → Y such that Ax ∈ φ(x), x ∈ X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces.