Hahn-Banach extension theorems

Abstract

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.