Locally Convex Spaces with the Extension Property

There are several researches on a normed space N with the extension property : each continuous linear function f on a subspace of any normed space with values in N has a linear extension f' on the whole space such that ‖f‖=‖f'‖. Among those, the following result has been obtained by L. Nachbin [4] ; a normed space has the extension property if and only if the collection of all its spheres has the binary intersection property. The collection U of all spheres is said to have the binary intersection property if every subcollection of U, any two elements of which intersect, has a nonvoid intersection. In this paper, we shall give a convenient definition of extension property of locally convex topological vector spaces. Of course the property must be a generalization of the usual extension property. Theorem 1 gives a necessary and sufficient condition in order that a locally convex space should have the extension property in our sense which corresponds with Nachbin's result. Theorem 2 gives a characterization of a locally convex topological vector space having the extension property. A vector space E is said to be a topological vector space if E is a Hausdorff space in which the vector operations, summation and scalar multiplication, are continuous for the topology. Moreover, if the neighbourhood system in a topological vector space consists of convex sets, then E is said to be a locally convex topological vector space or a locally convex space. In this paper, the neighbourhood system in a locally convex space is always assumed, without loss of generality, to consist of symmetric, convex closed sets. We shall denote by R the real space (-∞, +∞) and {x ; P(x)} the set of all the elements with the property P(x)