Characterization of Fr\'{e}chet Spaces and Application to Hausdorff MNC

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