Über die Gleichstetigkeit von Mengen von linearen Abbildungen und eine Klasse von topologischen linearen Räumen

In the first part of this paper we proof the following theorem: Let E and F be topological linear spaces, α an infinite cardinal number, and H a set of linear mappings from E into F such that every subset G of H with cardinality |G|≤α is equicontinuous. Then H is equicontinuous on every linear subspace of E which is the closed linear hull of a family (B_L;L∈I), |I|≤α, of precompact subsets of E. In the second part we introduce the class of all topological linear spaces E with the following property: A set H of linear mappings from E into a topological linear space is equicontinuous, if every countable subset of H is equicontinuous. We show that this class is closed with respect to forming topological products and linear final topologies.