Here we present an overview of countably-normed spaces. We discuss the main topologies—weak, strong, and inductive—placed on the dual of a countably-normed space and discuss the σ–fields generated by these topologies. In particular, we show that under certain conditions the strong and inductive topologies coincide and the σ–fields generated by the weak, strong, and inductive topologies are equa...

Here we present an overview of countably-normed spaces. We discuss the main topologies—weak, strong, and inductive—placed on the dual of a countably-normed space and discuss the σ–fields generated by these topologies. In particular, we show that under certain conditions the strong and inductive topologies coincide and the σ–fields generated by the weak, strong, and inductive topologies are equa...

Let X be a complex topological vector space and L(X) the set of all continuous linear operators on X. An operator T ? is supercyclic if there x such that, COrb(T,x) = {?Tnx : C, n 0}, dense in In this paper, we extend notion from single to subset L(X). We prove that most related proprieties supercyclicity case remains true for ?. This leads us obtain some results C-regularized groups operators.

Here we present an overview of countably-normed spaces. In particular, we discuss the main topologies—weak, strong, and inductive—placed on the dual of a countably-normed space and discuss the σ-fields generated by these topologies. The purpose in mind is to provide the background material for many of the results used is White Noise Analysis. 1. Topological Vector Spaces In the introduction we ...

In this paper we present new forms of the classical separation axioms on topological spaces. Our constructions generate a method to refine separation properties when passing to the quotient space and our results may be useful in the study of algebraic topological structures, such as topological groups and topological vector spaces. AMS subject classification: 54D10, 54D15, 54H11, 54H13.

Let X be a completely regular Hausdorff space, E a Hausdorff topological vector space, CL(E) the algebra of continuous operators on E, V a Nachbin family on X and F ⊆ CVb(X,E) a topological vector space (for a given topology). If π : X → CL(E) is a mapping and f ∈ F , let Mπ(f)(x) := π(x)f(x). In this paper we give necessary and sufficient conditions for the induced linear mapping Mπ to be a mu...