Topological and Functional Properties of Some F -Algebras of Holomorphic Functions

Let n p (1 < p < ∞) be the privalov class of holomorphic functions on the open unit disk d in the complex plane. the space n p equipped with the topology given by the metric d p defined by d p (f,g) = (∫ 0 2 π(l o g (1 + | f ∗ (e i θ) - g ∗ (e i θ) |)) p (d θ / 2 π)) 1 / p,f,g ε n p,becomes an f -algebra. for each p > 1,we also consider the countably normed fréchet algebra f p of holomorphic functions on d which is the fréchet envelope of the space n p. notice that the spaces f p and n p have the same topological duals. in this paper,we give a characterization of bounded subsets of the spaces f p and weakly bounded subsets of the spaces n p with p > 1. if (f p) ∗ denotes the strong dual space of f p and n p w ∗ denotes the space s p of complex sequences γ = { γ n } n satisfying the condition γ n = o e x p - c n 1 / (p + 1),equipped with the topology of uniform convergence on weakly bounded subsets of n p,then we prove that f p ∗ = n p w ∗ both set theoretically and topologically. we prove that for each p > 1 f p is a montel space and that both spaces f p and (f p) ∗ are reflexive. © 2015 romeo meštrović.