For any weightable quasi-metric space (X, d) having a maximum with respect to the associated order ≤d, the notion of the quasi-metric of complexity convergence on the the function space (equivalently, the space of sequences) X, is introduced and studied.We observe that its induced quasi-uniformity is finer than the quasi-uniformity of pointwise convergence and weaker than the quasi-uniformity o...

 The concept of statistical convergence in $2$-normed spaces for double sequence was introduced in [S. Sarabadan and S. Talebi, {it Statistical convergence of double sequences  in $2$-normed spaces }, Int. J. Contemp. Math. Sci. 6 (2011) 373--380]. In the first, we introduce concept strongly statistical convergence in $2$-normed spaces and generalize some results. Moreover,  we define the conce...

Set-valued mappings play an important role in a number of optimization problems. We examine several of those problems in these notes. We discuss iterative algorithms for solving these problems and prove convergence. We use the finite-dimensional space RJ through-out, although the results also hold for weak convergence in infinite-dimensional Hilbert space.

Nearness (a fuzzy nearness) is a fuzzy relation that can be used to model various grades of “being close” in a linear space. We study the uniform convergence of a sequence of functions with values in a space equipped with a nearness relation. The uniform convergence for the mappings into a space with a fuzzy nearness is defined and it is shown that a theorem similar to Moore-Osgood theorem for ...

The original motivation of this paper, as presented in [15], was to compute the dual space of a lattice of subsets of some free monoid A. According to StonePriestley duality, the dual space of a lattice can be identified with the set of its prime filters, but it is not always the simplest way to describe it. Consider for instance the Boolean algebra generated by the sets of the form uA, where u...

A metric compact space M is seen as the closure of the union of a sequence (Mn) of finite n-dense subsets. Extending to M (up to a vanishing uniform distance) Banach-space valued Lipschitz functions defined on Mn, or defining linear continuous near-extension operators for real-valued Lipschitz functions on Mn, uniformly on n is shown to be equivalent to the bounded approximation property for th...

In this paper we present a topology on the space of real-valued functions defined on a functionally Hausdorff space X that is finer than the topology of pointwise convergence and for which (1) the closure of the set of continuous functions C(X) is the set of upper semicontinuous functions on X , and (2) the pointwise convergence of a net in C(X) to an upper semicontinuous limit automatically en...

We study the accuracy of the discrete least-squares approximation on a finite dimensional space of a real-valued target function from noisy pointwise evaluations at independent random points distributed according to a given sampling probability measure. The convergence estimates are given in mean-square sense with respect to the sampling measure. The noise may be correlated with the location of...

Some Schur, Vitali-Hahn-Saks and Nikodým convergence theorems for (l)-group-valued measures are given in the context of (D)-convergence. We consider both the σ-additive and the finitely additive case. Here the notions of strong boundedness, countable additivity and absolute continuity are formulated not necessarily with respect to a same regulator, while the pointwise convergence of the measure...