Polynomials in linear relations

Abstract

The study of linear relations is continued in the setting of the theory of locally convex linear topological spaces. The investigation is limited to the polynomials in one fixed closed linear relation. Conditions both on the relation and on the locally convex space are discussed that are sufficient or necessary and sufficient for all the polynomials in this relation to be also closed. The reader is referred to [1] for full details of the algebraic properties of linear relations, and to [4] for a summary. Since we are concerned here with a special class of linear relations in locally convex spaces, we present a compendium of definitions tailored to this case. Let X be a locally convex space equipped with the Mackey-Arens topology. A linear relation in X is a linear subspace of X0X. This concept generalizes that of an operator on X. If T is a linear relation in X, the definitions of the domain and range of T, D(T) and R(T) respectively, are obvious. If S and T are linear relations in X,