Locally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space test functions into given locally lattice measures. The endowed with topology uniform convergence on bounded sets. Their structure is characterized via regularization and function-valued seminorms under mild structural assumptions Many recent gen...

Let E be a vector valued sequence space with operator valued βdual E . If E satisfies certain gliding hump assumptions, we show that pointwise bounded subsets of E are sequentially equicontinuous. The result is established by considering uniform convergence of the elements in E .

We study sample average approximations under adaptive importance sampling. Based on a Banach-space-valued martingale strong law of large numbers, we establish uniform convergence of the sample average approximation to the function being approximated. In the optimization context, we obtain convergence of the optimal value and optimal solutions of the sample average approximation.

Let X be a locally connected, b-compact metric space and E a closed subset of X. Let G be the space of all continuous real-valued functions defined on some closed subsets of E. We prove the equivalence of the τ aw and τ K topologies on G, where τ aw is the so called Attouch-Wets topology, defined in terms of uniform convergence of distance functionals, and τ K is the topology of Kuratowski conv...

We define a regularity axiom for lattice-valued convergence spaces where the lattice is a complete Heyting algebra. To this end, we generalize the characterization of regularity by a ”dual form” of a diagonal condition. We show that our axiom ensures that a regular T1-space is separated and that regularity is preserved under initial constructions. Further we present an extension theorem for a c...

In this study, we introduce the notion of pointwise and uniform statistical convergence of double sequences of real-valued functions. We also give the relations between these convergences and pointwise,uniform convergence. Furthermore, we introduce the concept of statistically Cauchy sequence and study statistical analogue of convergence and Cauchy criterion for double sequences of real-valued ...

The original definition of a topological space given by Hausdorff used neighborhood systems. Lattice-valued maps appear in this context when you identify a topology with a monoid in the Kleisli category of the filter monad on SET. H?hle’s notion of a lattice-valued topology [2] uses the same idea and it’s inspired in the classical lattice-valued topologies. Ltopological spaces are motivated by ...

This article generalises the concept of realised covariation to Hilbert-space-valued stochastic processes. More precisely, based on high-frequency functional data, we construct an estimator trace-class operator-valued integrated volatility process arising in general mild solutions Hilbert space-valued evolution equations sense Da Prato and Zabczyk (2014). We prove a weak law large numbers for t...

Based on a complete Heyting algebra, we modify the definition oflattice-valued fuzzifying convergence space using fuzzy inclusionorder and construct in this way a Cartesian-closed category, calledthe category of $L-$ordered fuzzifying convergence spaces, in whichthe category of $L-$fuzzifying topological spaces can be embedded.In addition, two new categories are introduced, which are called the...