Related papers: The Dvoretsky-Rogers Theorem for vector valued int…
A famous theorem due to Maurey and Pisier asserts that for an infinite dimensional Banach space $E$, the infumum of the $q$ such that the identity map $id_{E}$ is absolutely $\left( q,1\right) $-summing is precisely $\cot E$. In the same…
The distinction between conditional, unconditional, and absolute convergence in infinite-dimensional spaces has fundamental implications for computational algorithms. While these concepts coincide in finite dimensions, the Dvoretzky-Rogers…
In this paper, we deal with a notion of Banach space-valued mappings defined on a set consisting of finite graphs with uniformly bounded vertex degree. These functions will be endowed with certain boundedness and additivity criteria. We…
We establish the vector-valued Wiener type theorems for countable projective and inductive limits of quasi-Banach algebras in a weighted setting for both finite and infinite dimensional cases. As an application, we extend the notions of…
We introduce kernel-summability methods in Banach spaces using the vector-valued integrals and prove an analogue of the Silverman-Toeplitz Theorem for regular kernel-summability methods. We also show that if $X$ is a Banach space and one…
We extend the invertibility principle of J. Bourgain and L. Tzafriri to operators acting on arbitrary decompositions id = \sum x_j \otimes x_j, rather than on the coordinate one. The John's decomposition brings this result to the local…
The main result of the paper: Given any $\varepsilon>0$, every locally finite subset of $\ell_2$ admits a $(1+\varepsilon)$-bilipschitz embedding into an arbitrary infinite-dimensional Banach space. The result is based on two results which…
We prove a uniformly continuous linear extension principle in topological vector spaces from which we derive a very short and canonical construction of the Lebesgue integral of Banach space valued maps on a finite measure space. The Vitali…
Our main aim is to investigate the approximation properties for the summation integral type operators in a statistical sense. In this regard, we prove the statistical convergence theorem using well known Korovkin theorem and the degree of…
We formalize the observation that the same summability methods converge in a Banach space $X$ and its dual $X^*$. At the same time we determine conditions under which these methods converge in the weak and weak*-topologies on $X$ and $X^*$…
The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in…
We study when the integration maps of vector measures can be computed as pointwise limits of their finite rank Radon-Nikod\'ym derivatives. We will show that this can sometimes be done, but there are also principal cases in which this…
Uniqueness (up to isometries) and existence of limits are studied in the context of Cheeger-Gromov convergence of spacetimes. To address the non-compactness of the vector isometry group in the semi-Riemannian setting, standard pointed…
We define an integral of real-valued functions with respect to a measure that takes its values in the extended positive cone of a partially ordered vector space $E$. The monotone convergence theorem, Fatou's lemma, and the dominated…
In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continous convex functions on a vector space ${\bf R}^m$ and vector-valued functions in a weakly compact subset of a…
In classical analysis, the relationship between continuity and Riemann integrability is an intimate one: a continuous function on a closed and bounded interval is always Riemann integrable whereas a Riemann integrable function is continuous…
Extended real-valued functions are often used in optimization theory, but in different ways for infimum problems and for supremum problems. We present an approach to extended real-valued functions that works for all types of problems and…
We define a monad M on a category of measurable bornological sets, and we show how this monad gives rise to a theory of vector-valued integration that is related to the notion of Pettis integral. We show that an algebra X of this monad is a…
In this paper we study the problem of extending functions with values in a locally convex Hausdorff space $E$ over a field $\mathbb{K}$, which have weak extensions in a weighted Banach space $\mathcal{F}\nu(\Omega,\mathbb{K})$ of…
We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, `vector-valued' means `taking values in a function…