Algebras which represent their linear functionals

Abstract

This paper was originally intended to contain a generalization of a theorem of Banach on the extension of linear functionals. This generalized theorem now appears as a by-product of a study of a class of algebras which we believe to be of much greater interest than the theorem itself. Let X be a vector space over the real field and let π(x) be a sub-additive, positive-homogeneous functional on X. Banach ((2), pp. 27–9) proves that any real linear functional f on a subspace X0 of X which satisfies f(x) ≤ π(x) on X0 can be extended to a real linear functional F on X with F(x) ≤ π(x) on X. One of the essential differences between this theorem and the Hahn-Banach theorem is that π can take negative values.