Extending continuous linear functionals in convergence inductive limit spaces

Source abstract

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