A Theorem of the Hahn-banach Type for Linear Transformations

Every continuous linear functional defined on a vector subspace of a real normed space can be extended to the whole space so as to remain linear and continuous, and with the same norm(2). The extension of continuous linear transformations between two real normed spaces has been studied by several authors and for a long time it has been recognized that this problem has a close connection with the question of the existence of projections of norm one, and moreover that the nature of the space where the transformations take their values is much more important than that of the space where the transformations have to be defined. It is not known, as far as we can say, what are the precise conditions for the possibility of extending a transformation without disturbing its linearity, continuity, and norm. In this paper we shall give a necessary and sufficient condition, which refers only to the space where the transformation takes its values and does not involve the transformation itself or the space where it is to be defined, for such an extension to be possible: the condition is expressed in terms of a certain "binary intersection property" of the collection of spheres of the normed space (see Theorem 1). After that we proceed to the study of the structure of real normed spaces whose collections of spheres have this property. A first step in this direction is given by the theorem asserting that these normed spaces (provided they contain at least one extreme point in the unity sphere) are simply those that can be made into complete vector lattices with order unity in such a manner that the norm derived from the order relation and the order unity in the natural way is identical to the given norm (see Theorem 2, and Theorem 3 for the finite-dimensional case). In this connection we point out a conjecture which we have not been able to settle, namely that, if the collection of spheres of a normed space has the binary intersection property, then its unity sphere must contain an extreme point. By using some results of S. Kakutani and M. H. Stone, we establish the connection between the normed spaces having the binary intersection property and the spaces of real continuous functions over certain compact Hausdorff spaces (see Theorem 4), or the complete Boolean algebras (see Theorem 6).