Extending Continuous Linear Functionals in Convergence Inductive Limit Spaces

Abstract

Let ${E_n}$ be an increasing sequence of locally convex linear topological spaces such that the dual ${E’_n}$ of each has a Fréchet topology (not necessarily compatible with the dual system $({E’_n},{E_n}))$ weaker than the Mackey topology. Let $E = \bigcup \nolimits _{n = 1}^\infty {{E_n},F}$ be a subspace of $E$ and $\tau$ the inductive limit convergence structure on $E$. Conditions are given which insure that every $\tau$-continuous linear functional on $F$ has a $\tau$-continuous linear extension to $E$. This result generalizes a theorem of C. Foias and G. Marinescu.