Characterization of Fréchet Spaces and Application to Hausdorff MNC

Abstract

In this short note, we give a characterization of Fréchet spaces via properties of their metric. This allows us to prove that the Hausdorff measure of noncompactness (MNC), defined over Fréchet spaces, is indeed an MNC. As first applications, we lift well-known fixed-point theorems for contractive and condensing operators to the setting of Fréchet spaces.