Locally convex topologies and control theory

Complementation in the Lattice of Locally Convex Topologies

Tom Richmond* ([email protected]). Complementation in the Lattice of Locally Convex Topologies. We find all locally convex homogeneous topologies on (R,≤) and determine which of these have locally convex complements. Among the locally convex topologies on a n-point totally ordered set, each has a locally convex complement, and at least n of them have 2n−1 locally convex complements. For any ...

Locally Quasi-Convex Compatible Topologies on a Topological Group

For a locally quasi-convex topological abelian group (G, τ), we study the poset C (G, τ) of all locally quasi-convex topologies on G that are compatible with τ (i.e., have the same dual as (G, τ)) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G, Ĝ). Whether it has also a top element is an open question. We study both quantitative aspects of ...

Approximate Convexity and Submonotonicity in Locally Convex Spaces

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.

6 Smooth K - Theory of Locally Convex Algebras

Smooth K-functors are introduced and the smooth K-theory of locally convex algebras is developed. It is proved that the algebraic and smooth K-functors are isomorphic on the category of quasi ⊗̂-stable real (or complex) Fréchet algebras.