Convex Hausdorff Topological Vector Spaces Research Articles

Superdiagonal forms for completely continuous linear operators

Altman [1] showed that Riesz-Schauder theory remains valid for a completely continuous linear operator on a locally convex Hausdorflf topological vector space over the complex field. In a later paper [2], he proved an analogue of the Aronszajn-Smith result; specifically, he showed that such an operator possesses a proper closed invariant subspace. The purpose of this paper is to show that Ringrose's theory of superdiagonal forms for compact linear operators [3] can be generalized to the case of a completely continuous linear operator on a locally convex Hausdorff topological vector space over the complex field. However, the proof given in [3] requires considerable modification.

The facialQ-topology for compact convex sets

The structure of the set of closed two-sided ideals in aC*-algebraU with identity is described by means of a topology on the set ∂eK of extreme points of the state spaceK ofU. Recent results of Alfsen, Andersen, Combes, Perdrizet, Wils, and others have shown that such a topology can be defined on the set ∂eK of extreme points of an arbitrary compact convex subset of a locally convex Hausdorff topological vector space.

Sequential Compactness of X Implies a Completeness Property for C(X)

A locally convex Hausdorff topological vector space is said to be quasicomplete if closed bounded subsets of the space are complete, and von Neumann complete if closed totally bounded subsets are complete (equivalently, compact). Clearly quasi-completeness implies von Neumann completeness, and the converse holds in, for example, metrizable locally convex spaces. In this note we obtain a class of locally convex spaces for which the converse fails. Specifically, let X be a completely regular Hausdorff space, and let CC(X) denote the space of continuous real-valued functions on X, endowed with the compact-open topology.

On the monotone σ-envelope of A(K)

Let K be a compact convex set regularly embedded in a locally convex Hausdorff topological vector space E. Let Ab(K) be the GM-space (6) with unit 1K consisting of all real valued bounded of affine functions on K and let A(K) be the GM-subspace of Ab(K) consisting of continuous functions. For properties of K and A(K), see (2). The monotone σ-envelope A(K)μ of A(K) is the smallest subset of Ab(K) containing A(K) and such that the pointwise limit of every uniformly bounded monotone sequence (an) in A(K)μ also lies in A(K)μ. In (1), Alfsen shows that when K is a metrizable simplex A(K)μ is a vector lattice. In this note it is shown conversely that if K is metrizable and A(K)μ is a vector lattice, then K is a simplex.

A Universal Measure

It is shown that for any ring R of sets there is a locally convex Hausdorff topological vector space M(R) and a measure χ : R → M(R) such that for any other locally convex Hausdorff topological vector space W and any measure µ : R → W, there exists a unique continuous linear map µ : M(R) → W such that . Measure is defined in such a way that M(R)′, the space of all continuous linear functionals on M(R), corresponds to all finite, signed measures in the usual sense when R is a σ‐ring. The problem of extending measures from R to the σ‐ring generated by R is formulted in this setting. Properties of the universal measure χ are described. The topology of M(R) is studied via the pairing <M(R), M(R)′>. For example, the weak compact subsets of M(R)′ are characterized. This characterization gives rise to a simple proof of the Orlicz‐Pettis theorem.

Extended Schauder decompositions of locally convex spaces

Let E[τ] be a locally convex Hausdorff topological vector space. An extended decomposition of E[τ] is a family {Ea}α∈A of closed subspaces of E such that, for each x in E and each α in A, there exists a unique point xα in Eα, with Here convergence will have the following meaning. Let Ф denote the set of all finite subsets of A. The sum is said to be convergent to x if for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф containing φ0. It follows that is Cauchy if and only if, for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф disjoint from φ0.

Measurability of Lattice Operations in a Cone

Let $X$ be a locally convex Hausdorff topological vector space and $C$ a convex cone generating $X$ such that $C$ is a lattice in its own order. Under suitable conditions $(x,y) \to \sup (x,y)$ and $\inf (x,y)$ are shown to be measurable mappings.

Measurability of lattice operations in a cone

Let $X$ be a locally convex Hausdorff topological vector space and $C$ a convex cone generating $X$ such that $C$ is a lattice in its own order. Under suitable conditions $(x,y) \to \sup (x,y)$ and $\inf (x,y)$ are shown to be measurable mappings.

Regularity of group valued Baire and Borel measures

It is known that a real valued measure (1) on the a-ring of Baire sets of a locally compact Hausdorff space, or (2) on the Borel sets of a complete separable metric space is regular. Recently Dinculeanu and Kluvanek used regularity of nonnegative Baire measures to prove that any Baire measure with valies in a locally convex Hausdorff topological vector space (TVS) is regular. Subsequently a direct proof of the same result was offered by Dinculeanu and Lewis. Here we show just as directly that any measure defined as in (1) or (2) is regular, even when it takes values in a Hausdorff topological group. In particular, when the group is a Hausdorff TVS, our result improves the Dinculeanu-KluvanekLewis theorem.

A Riesz representation theorem

Introduction. This paper deals with a Riesz representation theorem in the following setting: Let H be a compact Hausdorff space, let E and be locally convex Hausdorff topological vector spaces over the real or complex field. Let C(H, E) denote the continuous functions from H into E with the topology of uniform convergence. The purpose of this paper is to improve certain integral representation theorems for continuous linear transformations T from C(H, E) into The well-known Riesz representation theorem [10] gives a Stieltjes integral representation for T when H is a closed interval and when E and are the real numbers. There have been many generalizations of this theorem in the literature, and there have been two essentially different approaches giving rise to two different kinds of representation theorems. In one approach (see [1], [2], [5], [8], [9], [11]) y'T is written as an integral where y' is in the topological dual of The integral converges in the weak topology with a measure defined on the Borel sets with values in L(E, F), the space on continuous linear transformations of E into F. Under various additional assumptions on E, F, or T, T is written as an integral. (For example, this can be done if T is assumed to be compact or is assumed to be reflexive.) The most general of these theorems is due to Swong [11, Theorem 6.1, p. 283]. Another approach in the literature can be found in the papers [6], [12], [13], [14]. In these papers the method is of a constructive nature and T is written as an integral, and the convergence of the integral is in the e00 topology of F (norm topology in the case is a normed space), and T is thought of as a mapping of E into F by the canonical imbedding of into F. This is done by restricting the class of sets the measure is defined on. In [12] the class of sets is the intervals where H = [0, I], and E and are assumed to be normed spaces. In [6] and [13] the class of sets is the ring generated by compact Gj's. Recently in [14] Tucker and Wayment have been able to extend the measure to a larger proper subring of the Baire sets. In this paper it is shown that the restriction of the measure in [61, [12], [13], [14] was not necessary, and in fact T can be represented as an integral with convergence in the e topology with the measure defined on the Borel sets. This result thus implies and strengthens

Sur les programmes convexes définis dans des espaces vectoriels topologiques

We study optimally conditions and duality for convex programs : sup {f(x)|g(x)≤0} where f is a concave real-valued function which is defined on a subset of a real locally convex Hausdorff topological vector space E, and where g is a convex mapping of a subset of E into a real locally convex Hausdorff ordered topological vector space G. For this purpose, we define subdifferentials and a conjugate function of a G-valued function. We also introduce locally convex subsets and locally convex functions defined on a locally convex topological vector space. We obtain, in this way, several extensions of well-known results in convex programming.

Topological degree of non-compact mappings

AbstractWe extend the notion of topological degree to a suitable class of continuous mappings of a locally convex Hausdorff topological vector space into itself which are not completely continuous. This extension and the resulting existence theorems for non-linear equations are further extended to product spaces. Our extension is developed with application to quantum field theory in mind; this application is made elsewhere.

Σ-symmetric locally convex spaces

In [1] it is shown that barrelledness and quasi-barrelledness are merely the two extreme examples of a property, called 2-symmetry, which may be possessed by a locally convex Hausdorff topological vector space. The object of this note is to show how recent characterisations [2; 3] of barrelled and quasi-barrelled spaces may be subsumed under characterisations of I-symmetric spaces, and to exhibit some properties of these spaces. First we need some definitions and simple results.