Topological and Functional Properties of Some F-Algebras of Holomorphic Functions. (17th September 2015) - Meštrović, Romeo
Abstract : LetN p ( 1 < p < ∞ ) be the Privalov class of holomorphic functions on the open unit diskD in the complex plane. The spaceN p equipped with the topology given by the metricd p defined byd 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 ∈ N p, becomes anF -algebra. For eachp > 1, we also consider the countably normed Fréchet algebraF p of holomorphic functions onD which is the Fréchet envelope of the spaceN p . Notice that the spacesF p andN p have the same topological duals. In this paper, we give a characterization of bounded subsets of the spacesF p and weakly bounded subsets of the spacesN p withp > 1 . If( F p ) ∗ denotes the strong dual space ofF p andN p w ∗ denotes the spaceS p of complex sequencesγ = γ n n satisfying the conditionγ n = O e x p - c n 1 / ( p + 1 ), equipped with the topology of uniform convergence on weakly bounded subsets ofN p, then we prove thatF p ∗ = N p w ∗ both set theoretically and topologically. We prove that for eachp > 1 F p is a Montel space and that both spacesF p and( F p ) ∗ are reflexive.