Search results for "Topological vector space"

Dual attachment pairs in categorically-algebraic topology

2011

[EN] The paper is a continuation of our study on developing a new approach to (lattice-valued) topological structures, which relies on category theory and universal algebra, and which is called categorically-algebraic (catalg) topology. The new framework is used to build a topological setting, based in a catalg extension of the set-theoretic membership relation "e" called dual attachment, thereby dualizing the notion of attachment introduced by the authors earlier. Following the recent interest of the fuzzy community in topological systems of S. Vickers, we clarify completely relationships between these structures and (dual) attachment, showing that unlike the former, the latter have no inh…

On the Almost Everywhere Convergence of Multiple Fourier-Haar Series

2019

The paper deals with the question of convergence of multiple Fourier-Haar series with partial sums taken over homothetic copies of a given convex bounded set $$W\subset\mathbb{R}_+^n$$ containing the intersection of some neighborhood of the origin with $$\mathbb{R}_+^n$$ . It is proved that for this type sets W with symmetric structure it is guaranteed almost everywhere convergence of Fourier-Haar series of any function from the class L(ln+L)n−1.

Group topologies coarser than the Isbell topology

2011

Abstract The Isbell, compact-open and point-open topologies on the set C ( X , R ) of continuous real-valued maps can be represented as the dual topologies with respect to some collections α ( X ) of compact families of open subsets of a topological space X . Those α ( X ) for which addition is jointly continuous at the zero function in C α ( X , R ) are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections α ( X ) for which C α ( X , R ) is a topological vector space are defined canonically. The Isbell topology coincides with this vector space topology if and only if X is infraconsonant. Examples based on measure theoretic methods, t…

Fuzzy algebras as a framework for fuzzy topology

2011

The paper introduces a variety-based version of the notion of the (L,M)-fuzzy topological space of Kubiak and Sostak and embeds the respective category into a suitable modification of the category of topological systems of Vickers. The new concepts provide a common framework for different approaches to fuzzy topology and topological systems existing in the literature, paving the way for studying the problem of interweaving algebra and topology in mathematics, which was raised by Denniston, Melton and Rodabaugh in their recent research on variable-basis topological systems over the category of locales.

Topological Dual Systems for Spaces of Vector Measure p-Integrable Functions

2016

[EN] We show a Dvoretzky-Rogers type theorem for the adapted version of the q-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector value…

Remarks on the semivariation of vector measures with respect to Banach spaces.

2007

Suppose that and . It is shown that any Lp(µ)-valued measure has finite L2(v)-semivariation with respect to the tensor norm for 1 ≤ p < ∞ and finite Lq(v)-semivariation with respect to the tensor norm whenever either q = 2 and 1 ≤ p ≤ 2 or q > max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor norm for any 1 ≤ q < 2. It is also shown that the measure m (A) = χA has infinite Lq-semivariation with respect to the tensor norm if q < p.

On Certain Metrizable Locally Convex Spaces

1986

Publisher Summary This chapter discusses on certain metrizable locally convex spaces. The linear spaces used are defined over the field IK of real or complex numbers. The word "space" will mean "Hausdorff locally convex space". This chapter presents a proposition which states if U be a neighborhood of the origin in a space E. If A is a barrel in E which is not a neighborhood of the origin and F is a closed subspace of finite codimension in E’ [σ(E’,E)], then U° ∩ F does not contain A° ∩ F. Suppose that U° ∩ F contain A° ∩ F. Then A° ∩ F is equicontinuous hence W is also equicontinuous. Since W° is contained in A, it follows that A is a neighborhood of the origin, a contradiction.

K-theory of function rings

1990

AbstractThe ring R of continuous functions on a compact topological space Xwith values in R or C is considered. It is shown that the algebraic K-theory of such rings with coefficients in ZkZ, k any positive integer, agrees with the topological K-theory of the underlying space X with the same coefficient rings. The proof is based on the result that the map from Rδ (R with discrete topology) to R (R with compact-open topology) induces a natural isomorphism between the homologies with coefficients in ZkZ of the classifying spaces of the respective infinite general linear groups. Some remarks on the situation with X not compact are added.

Normed vector spaces consisting of classes of convex sets

1965

A remark on weakly convex continuous mappings in topological linear spaces

2009

Abstract Let C be a compact convex subset of a Hausdorff topological linear space and T : C → C a continuous mapping. We characterize those mappings T for which T ( C ) is convexly totally bounded.