In a first course in functional analysis, a great deal of time is spent with Banach spaces, especially the interaction between such spaces and their dual spaces. Banach spaces are a special type of topological vector space, and there are important topological vector spaces which do not lie in the Banach category, such as the Schwartz spaces. The most fundamental theorem about Banach spaces is t...

This paper deenes for each object X that can be constructed out of a nite number of vertices and cells a vector fX lying in a nite dimensional vector space. This is the ag vector of X. It is hoped that the quantum topological invariants of a manifold M can be expressed as linear functions of the ag vector of the i-graph that arises from any suitable triangulation T of M. Flag vectors are also d...

This paper defines for each object X that can be constructed out of a finite number of vertices and cells a vector f X lying in a finite dimensional vector space. This is the flag vector of X. It is hoped that the quantum topological invariants of a manifold M can be expressed as linear functions of the flag vector of the i-graph that arises from any suitable triangulation T of M. Flag vectors ...

Let a topological space. An intersection graph on a topological space , which denoted by ‎ , is an undirected graph which whose vertices are open subsets of and two vertices are adjacent if the intersection of them are nonempty. In this paper, the relation between topological properties of  and graph properties of ‎  are investigated. Also some classifications and representations for the graph ...

in this article we introduce the concept of $z^circ$-filter on a topological space $x$. we study and investigate the behavior of $z^circ$-filters and compare them  with corresponding ideals, namely, $z^circ$-ideals of $c(x)$,  the ring of real-valued continuous functions on a completely regular hausdorff space $x$. it is observed that $x$ is a compact space if and only if every $z^circ$-filter ...

in this work, we introduce notion of connectedness on fuzzy soft topological spaces and present fundamentals properties. we also investigate effect to fuzzy soft connectedness. moreover, c_i-connectedness which plays an important role in fuzzy topological space extend to fuzzy soft topological spaces.

in this paper we introduce the concept of topological number for locally convex topological spaces and prove some of its properties. it gives some criterions to study locally convex topological spaces in a discrete approach.

These notes give a summary of results that everyone who does work in functional analysis should know about the weak topology on locally convex topological vector spaces and the weak-* topology on their dual spaces. The most striking of the results we prove is Theorem 9, which shows that a subset of a locally convex space is bounded if and only if it is weakly bounded. It is straightforward to p...

The line density of line defects in terms of a two-component vector order parameter are obtained from the definition of topological charges of line defects. The spatial structure and bifurcation of line defects in three-dimensional space are also studied from the topological properties of the two-component vector order parameter. The branch conditions for generating, annihilating, colliding, sp...

In this paper, we consider strong form of a vector equilibrium problem and establish some existence results for solutions of such problem in the setting of topological vector spaces. We provide several coercivity conditions under which strong vector equilibrium problem has a solution. Our results generalize and extend the results of Bianchi and Pini [Coercivity conditions for equilibrium proble...