R. Beattie

Continuous Duality of Limits and Colimits of Topological Abelian Groups

In the category Cgp of convergence groups, the continuous dual Γc( ·) is a left adjoint and takes colimits to limits in Cgp. In general, limits are not taken to colimits. In this paper we show that, if we restrict ourselves to limits of topological groups, then reduced projective limits are carried to inductive limits in Cgp. As a consequence of this we show that the inductive limit in Cgp of locally compact topological groups is reflexive if it is separated.

Uniform convergence spaces

Uniform continuity, completeness and equicontinuity are the most important features of uniformities and uniform spaces. Uniform convergence spaces, the convergence generalization of uniform spaces, are not as strong as their topological counterparts. In particular uniform continuity is not a very strong property. But basically all properties of completeness can be carried over to uniform convergence spaces and equicontinuity is an even stronger concept in this more general setting since it is a part of the Arzela-Ascoli theorem. This theorem, which characterizes relative compactness in function spaces endowed with the continuous convergence structure, has far reaching applications.

The closed graph theorem

Closed graph theorems give sufficient conditions to guarantee that a linear mapping with a closed graph is continuous. Time has established this result as one of the fundamental principles of functional analysis. The first version was due to Banach [Ba32] and took place in the setting of Frechet spaces. This theorem proved to be so useful that great efforts were made over the next decades to increase its scope: to enlarge the classes of spaces which could act as domain spaces and codomain spaces for a closed graph theorem.

Continuous convergence and functional analysis

Abstract The usual setting for Functional Analysis is the category LCS of locally convex topological vector spaces. There are, however, advantages in working in a larger setting, the category CVS of convergence vector spaces—even if ones interest is restricted to LCS . In CVS , one has access to a dual structure, continuous convergence, unavailable in LCS . We show that theorems such as Grothendiecks completion theorem, Ptaks closed graph and open mapping theorems and the Banach-Steinhaus theorem are transformed from technical results in LCS to transparent and elegant results when examined in CVS with continuous convergence. In the theory of distributions, important bilinear mappings such as evaluations, multiplication and convolution, which are separately continuous when viewed in LCS , become jointly continuous in CVS .

Countability, Completeness and the Closed Graph Theorem

The webs of M. De Wilde [4] have made an enormous contribution to the closed graph theorems in locally convex spaces(lcs). Although webs have a very intricate layered construction, two properties in particular have contributed to the closed graph theorem. First of all, webs possess a strong countability condition in the range space which suitably matches the Baire property of Frechet spaces in the domain space; as a result the zero neighbourhood filter is mapped to a p-Cauchy filter, a filter attempting to settle down. Secondly webs provide a completeness condition which allow p-Cauchy filters to converge.

The Banach-Steinhaus theorem

While the content of the classical Banach-Steinhaus theorem varies somewhat in the literature, one very common variation is the following: if E and F are locally convex topological vector spaces and E is barrelled, then every pointwise bounded subset of 𝓛(E, F) is equicontinuous. This powerful theorem is used, for example to show that the pointwise limit of a sequence of continuous linear mappings is a continuous linear mapping. It is used as well to derive the continuity of separately continuous bilinear mappings.

Convergence vector spaces

The notion of convergence vector space arose in 1.2.8. Since these spaces are the main topic of this book, we examine their properties in some detail in this chapter.

Hahn-Banach extension theorems

The Hahn-Banach problem for convergence vector spaces has its roots in classical functional analysis. Let E be a strict topological 𝓛F-space, M a vector subspace of E with the property that M ∩E n is closed in each E n - such a subspace is called stepwise closed. Further, let φ bea sequentially continuous linear functional on M. Does there exist a (sequentially) continuous linear extension to E? This is a difficult and much researched problem. Subspaces with the property that all sequentially continuous linear functionals have a continuous linear extension have been called well-located in the literature. In Section 5 we show how such spaces are related to the question of the solution of partial differential equations.