Bornological universes were introduced by Hu in [11] and obtained renewed interest in recent articles on convergence in hyperspaces and function spaces and optimization theory. In [11] and [12] Hu gives a necessary and sufficient condition for which a bornological universe is metrizable. In this article we will give a characterization of uniformizable bornological universes. Furthermore, a cons...

In this paper, firstly, we obtain some new results about bornological convergence in locally convex cones (which was studied in [1]) and then we introduce the concept of bornological completion for locally convex cones. Also, we prove that the completion of a bornological locally convex cone is bornological. We illustrate the main result by an example.

The space of entire functions represented by Dirichlet series of several complex variables has been studied by S. Dauod [1]. M.D. Patwardhan [6] studied the bornological properties of the space of entire functions represented by power series. In this work we study the bornological aspect of the space Γ of entire functions represented by Dirichlet series of several complex variables. By Γ we den...

We define the ε∞-product of a Banach space G by a quotient bornological space E | F that we denote by Gε∞(E | F ), and we prove that G is an L∞-space if and only if the quotient bornological spaces Gε∞(E | F ) and (GεE) | (GεF ) are isomorphic. Also, we show that the functor .ε∞. : Ban× qBan −→ qBan is left exact. Finally, we define the ε∞-product of a b-space by a quotient bornological space a...

We introduce and study the concept of a bornological quantum group. This generalizes the theory of algebraic quantum groups in the sense of van Daele from the algebraic setting to the framework of bornological vector spaces. Working with bornological vector spaces, the scope of the latter theory can be extended considerably. In particular, the bornological theory covers smooth convolution algeb...

We define equivariant periodic cyclic homology for bornological quantum groups. Generalizing corresponding results from the group case, we show that the theory is homotopy invariant, stable and satisfies excision in both variables. Along the way we prove Radfords formula for the antipode of a bornological quantum group. Moreover we discuss anti-Yetter-Drinfeld modules and establish an analogue ...

The aim of this paper is that of discussing closed graph theorems for bornological vector spaces in a self-contained way, hoping to make the subject more accessible to non-experts. We will see how to easily adapt classical arguments of functional analysis over R and C to deduce closed graph theorems for bornological vector spaces over any complete, non-trivially valued field, hence encompassing...

We establish the existence of a stochastic integral in a nuclear space setting as follows. Let E, F, and G be nuclear spaces which satisfy the following conditions: the spaces are reflexive, complete, bornological spaces such that their strong duals also satisfy these conditions. Assume that there is a continuous bilinear mapping of E × F into G. If H is an integrable, E-valued predictable proc...

In this paper, we show that the category of Mackey-complete, separated, topological convex bornological vector spaces and bounded linear maps is a differential category. Such spaces were introduced by Frölicher and Kriegl, where they were called convenient vector spaces. While much of the structure necessary to demonstrate this observation is already contained in Frölicher and Kriegl’s book, we...