OnF-AlgebrasMp (1

Abstract

We consider the classesMp (1<p<∞)of holomorphic functions on the open unit disk𝔻in the complex plane. These classes are in fact generalizations of the classMintroduced by Kim (1986). The spaceMpequipped with the topology given by the metricρpdefined byρp(f,g)=f-gp=∫02π‍logp1+Mf-gθdθ/2π1/p, withf,g∈MpandMfθ=sup0⩽r<1⁡f(reiθ), becomes anF-space. By a result of Stoll (1977), the Privalov spaceNp (1<p<∞)with the topology given by the Stoll metricdpis anF-algebra. By using these two facts, we prove that the spacesMpandNpcoincide and have the same topological structure. Consequently, we describe a general form of continuous linear functionals onMp(with respect to the metricρp). Furthermore, we give a characterization of bounded subsets of the spacesMp. Moreover, we give the examples of bounded subsets ofMpthat are not relatively compact.