On spaces of vector-valued functions modeled on the Hardy and Bergman algebras

The strong N p ( X ) , N p ( X ) and weak w N p ( X ) , w N p ( X ) spaces of analytic functions with values in a Banach space X , modeled on the large Hardy N p and Bergman N p algebras respectively, are studied. The Fréchet envelopes and the continuous linear functionals on these spaces are described. We show that, although that strong and weak spaces are different, they have the same locally convex structure (the Fréchet envelopes are equal).