Vector-valued holomorphic functions revisited

Let Omega subset of C be open, X a Banach space and W subset of X'. We show that every sigma(X, W)-holomorphic function f : Omega --> X is holomorphic if and only if every sigma(X, W)-bounded set in X is bounded. Things are d ifferent if we assume f to be locally bounded. Then we show that it suffice s that phi circle f is holomorphic for all phi is an element of W, where W is a separating subspace of X' to deduce that f is holomorphic. Boundary Ta uberian convergence and membership theorems are proved. Namely, if boundary values tin a weak sense) of a sequence of holomorphic functions converge/b elong to a closed subspace on a subset of the boundary having positive Lebe sgue measure, then the same is true for the interior points of Omega, unifo rmly on compact subsets. Some extra global majorants are requested. These r esults depend on a distance Jensen inequality. Several examples are provide d (bounded and compact operators; Toeplitz and Hankel operators; Fourier mu ltipliers and small multipliers).