ABSTRACT
Let E and F be complex Banach spaces. To each compact subset K of E′ we associate a seminorm on E defined by pK (x) := sup{|φ{x)| : φ ∈ K}. Let Eθ be the space E endowed by the topology generated by the seminorms p K when K runs over the compact subsets of E′. We consider the space Hu (Eθ ; F) of all the f : E → F that are holomorphic in {E, p K ) for some K and show that Hu {Eθ ; F) = H(Eθ ; F) if E′ is separable. We also study the spaces Hu (Eθ ; F) and H(Eθ ; F) in connection with the spaces Hwu (E; F) and Hw (E; F).
