On Vector Integral Inequalities

2007

A construction of product measures in complete bornological locally convex topological vector spaces is given. Two theorems on the existence of the bornological product measure are proved. A Fubini-type theorem is given. Mathematics Subject Classification 2000: Primary 46G10, Secondary 28B05

2019

Let X, Y be Banach spaces (or either topological vector spaces) and let us consider the function space C (S, X) of all continuous functions f: S → X, from the compact (locally compact) space S into X, equipped with some appropriate topology. Put C (S, X) = C (S) if X = R. In this work we will mainly be concerned with the problem of representing linear bounded operators T: C (S, X) → Y in an integral form: f ∈ C (S, X), Tf =R S f dµ, for some integration process with respect to a measure µ on the Borel σ−field BS of S. The prototype of such representation is the theorem of F. Riesz according to which every continuous functional T: C (S) → R has the Lebesgue integral form Tf =R S f dµ. This paper is intended to present various extensions of this theorem to the Banach spaces setting alluded to above, and to the context of locally convex spaces.

We review the development of the theory of integra- tion with respect to a vector measure with values in a Banach space. The starting point is a 1955 paper by Bartle, Dunford and Schwartz where the authors consider the vector version of Riesz's Theorem on bounded linear functionals on spaces of continuous functions over a com- pact space. Next we address the quest for the right look at the space of such integrable functions. We end by looking at applications of the theory.

Advances in Mathematics: Scientific Journal

This paper addresses some properties of vector measures (Banach space valued measures) as well as topological results on some spaces of vector measures of bounded variation.

Indagationes Mathematicae, 1997

For a real Frtchet space X with dual X' the following conditions are equivalent: (a) X admits a continuous norm. (b) Every convex and weakly compact subset of X is the closed convex hull of its exposed points. (c) For every X-valued. countably additive measure v there exists I' in .Y' such that I, is ~.\-'IJ~continuous.

2010

The bornological product measures via the generalized Dobrakov integral in complete bornological locally convex spaces are studied using the domination of considered vector measures. A Fubini-type theorem for such product measures is proven.

Journal of Applied Analysis, 2007

The solvability of the generalized weak vector implicit variational inequality problem, generalized strong vector implicit variational inequality problem and generalized vector variational inequality problem are proved by using a generalized Fan's KKM theorem. Our results extend and unify corresponding results of other authors.

Applied Mathematics Letters, 1997

In this note, we first prove existence theorems for noncompact generalized quasivariational inequalities. As applications, two fixed point theorems for upper or lower semicontinuous multivalued mappings without compact domains are given in locally Hausdorff topological vector spaces. These results generalize or improve corresponding results in the literature.

Positivity, 2008

Let ν be a vector measure with values in a Banach space Z. The integration map Iν : L 1 (ν) → Z, given by f → f dν for f ∈ L 1 (ν), always has a formal extension to its bidual operator I * * ν : L 1 (ν) * * → Z * *. So, we may consider the "integral" of any element f * * of L 1 (ν) * * as I * * ν (f * *). Our aim is to identify when these integrals lie in more tractable subspaces Y of Z * *. For Z a Banach function space X, we consider this question when Y is any one of the subspaces of X * * given by the corresponding identifications of X, X (the Köthe bidual of X) and X * (the topological dual of the Köthe dual of X). Also, we consider certain kernel operators T and study the extended operator I * * ν for the particular vector measure ν defined by ν(A) := T (χA).

Quaestiones Mathematicae, 1997

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 Lusin-measurable 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 Λ {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus 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.

Demonstratio Mathematica, 1988

Journal of Mathematical Extension, 2017

We extend the notions of integration and differentiation to cover the class of functions taking values in topological vector spaces. We give versions of the Lebesgue-Nikodym Theorem and the Fundamental Theorem of Calculus in such a more general setting.

Mathematische Nachrichten, 1995

We study the structure of bounded sets in the space L ' ( 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:L' -. E, a fact well-known for Banach spaces. We also study the relationship between Radon-Nikodym property and the characterization of the dual of L'{E} as the space LODIE;}. R

Advances in Pure Mathematics, 2013

We investigate the H-stochastic integral introduced in [24]. It is known that this integral generalizes the classical It^o stochastic integral and the It^o integral on a Fock space. In the present paper we construct and study an extension of the H-stochastic integral which will generalize the Hitsuda{Skorokhod integral.