On F-Algebras Mp (1

Abstract EN

We consider the classes Mp (1

These classes are in fact generalizations of the class M introduced by Kim (1986).

The space Mp equipped with the topology given by the metric ρp defined by ρp(f,g)=f-gp=∫02πlogp1+Mf-gθdθ/2π1/p, with f,g∈Mp and Mfθ=sup0⩽r<1f(reiθ), becomes an F-space.

By a result of Stoll (1977), the Privalov space Np (1

By using these two facts, we prove that the spaces Mp and Np coincide and have the same topological structure.

Consequently, we describe a general form of continuous linear functionals on Mp (with respect to the metric ρp).

Furthermore, we give a characterization of bounded subsets of the spaces Mp.

Moreover, we give the examples of bounded subsets of Mp that are not relatively compact.