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This paper presents a brief survey of the theory of stochastic integration in Banach spaces. Expositions of the stochastic integrals in martingale type 2 spaces and UMD spaces are presented, as well as some applications of the latter to…
The purpose of this article is to present the construction and basic properties of the general Bochner integral. The approach presented here is based on the ideas from the book The Bochner Integral by J. Mikusinski where the integral is…
This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex…
This paper is devoted to a systematic study and characterizations of the fundamental notions of variational and strong variational convexity for lower semicontinuous functions. While these notions have been quite recently introduced by…
Brownian motions, martingales, and Wiener processes are introduced and studied for set valued functions taking values in the subfamily of compact convex subsets of arbitrary Banach space $X$. The present paper is an application of one the…
The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional)…
Banach spaces that are complemented in the second dual are characterised precisely as those spaces $X$ which enjoy the property that for every amenable semigroup $S$ there exists an $X$-valued analogue of an invariant mean defined on the…
We study dentable maps from a closed convex subset of a Banach space into a metric space as an attempt of generalize the Radon-Nikod\'ym property to a "less linear" frame. We note that a certain part of the theory can be developed in rather…
We study functions of bounded variation (and sets of finite perimeter) on a convex open set $\Omega\subseteq X$, $X$ being an infinite dimensional real Hilbert space. We relate the total variation of such functions, defined through an…
In this paper, we introduce new properties of the relative interior calculus for nearly convex sets, functions, and set-valued mappings. These properties are important for the development of duality theory in optimization. Then we…
In the paper we introduce Orlicz type functional spaces defined in terms of nonlocal convolution type integral functionals and study the main properties of these spaces. We show in particular that, under natural convexity and growth…
This paper considers a general framework for the study of the existence of quasi-variational and variational solutions to a class of nonlinear evolution systems in convex sets of Banach spaces describing constraints on a linear combination…
In this work, a novel approach for the solution of the inverse conductivity problem from one and multiple boundary measurements has been developed on the basis of the implication of the framework of BV - functions. The space of the…
We introduce a sheaf theoretic viewpoint on functional analysis designed for infinite dimensional Lie group actions. We develop functional calculus for Banach valued functors and, in particular, prove the existence of an exponential map for…
We consider abstract Banach spaces of analytic functions on general bounded domains that satisfy only a minimum number of axioms. We describe all invertible (equivalently, surjective) weighted composition operators acting on such spaces.…
The paper deals with (multidimensional and one-dimensional) Bochner-Phillips functional calculus. Bounded perturbations of Bernstein functions of (one or several commuting) semigroup generators on Banach spaces are considered, conditions…
In an M-type 2 Banach space, firstly we explore some properties of the set-valued stochastic integral associated with the stationary Poisson point process. By using the Hahn decomposition theorem and bounded linear functional, we obtain the…
We study the existence of global implicit functions for equations defined on open subsets of Banach spaces. The partial derivative with respect to the second variable is only required to have a left inverse instead of being invertible.…
This article is devoted to the stability of error bounds (local and global) for semi-infinite convex constraint systems in Banach spaces. We provide primal characterizations of the stability of local and global error bounds when systems are…
In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of…