Weak locally multiplicatively-convex algebra

A class of certain properties of approximately n-multiplicative maps between locally multiplicatively convex algebras

‎We extend the notion of approximately multiplicative to approximately n-multiplicative maps between locally multiplicatively convex algebras and study some properties of these maps‎. ‎‎W‎e prove that every approximately n-multiplicative linear functional on a functionally continuous locally multiplicatively convex algebra is continuous‎. ‎We also study the relationship between approximately mu...

Application of Jacobi’s Representation Theorem to Locally Multiplicatively Convex Topological R-algebras

Let A be a commutative unital R-algebra and let ρ be a seminorm on A which satisfies ρ(ab) ≤ ρ(a)ρ(b). We apply T. Jacobi’s representation theorem [10] to determine the closure of a ∑A-module S of A in the topology induced by ρ, for any integer d ≥ 1. We show that this closure is exactly the set of all elements a ∈ A such that α(a) ≥ 0 for every ρ-continuous R-algebra homomorphism α ∶ AÐ→ R wit...

Bornological Completion of Locally Convex Cones

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 weak topology of locally convex spaces and the weak-* topology of their duals

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...

Some Results on Boundedness in Locally Convex Cones