Rings of Operators on Vector Spaces

Abstract

Eidelheit' has shown that the algebraic properties of the ring of all bounded linear transformations of a real Banach space into itself characterize the space up to an isomorphism. Mackey [19, 201 has extended this result to a more general class of real spaces. It is the purpose of this paper to examine more closely the relationship between ring and space and to extend the results, so far as possible, to complex spaces. We consider a vector space V together with a total2 family P of linear3 functionals defined on V. A subset S of V is said to be bounded if each of the functionals in P is bounded on S. Throughout the paper, unless otherwise stated, the word space is used to refer to a complex vector space in which a notion of bounded set has been introduced by this means. The conjugate space V* of V is defined, as for Banach spaces, to be the vector space of all bounded linear functionals defined on V. The family Q of functionals defined on V* which is used to derive the notion of bounded set in V* is taken to be'