A NECESSARY CONDITION FOR A DUGUNDJI EXTENSION PROPERTY

The paper is organized into three sections. In Section 1 we utilize a proof by E. Michael of the Arens-Eells Embedding Theorem to show that if every continuous function from a subspace S of a topological space X into a complete locally convex topological vector space extends to X, then (X,S) satisfies a property relating to the simultaneous order-preserv ing extension of the bounded continuous pseudometrics on S. Section 2 introduces various related 'Dugundji-type' extension properties. Results are given showing the relationship of these properties to various concepts involving the simultaneous linear extension of functions. Section 3 consists of examples. Section 1 The Arens-Eells Embedding Theorem [2] states that every metric space (X,d) can be embedded isometrically as a closed, linearly independent subset of a normed linear space. Michael's proof [12] involves choosing a metric space (Y,d) containing (X,d) and with a point Yo E Y-X. Lip(Y) = {f: Y ~ R: f(y) = 0 and for some K> 0, If(x)- f(y) I < a K d(x,y) for all x,y E y}. With Ilf II equal to the infimum of the K's that work, Lip(Y) becomes a Banach space, and its dual E is also a Banach space. The map h: X ~ E is defined by h(x) (f) = f(x) for all f E Lip(Y). It is then shown that h is an isometry and if xl,···,x are distinct elements of X, n 266 Sennott then {h(x) , •• •,h(x)} forms a linearly independent set inl n E. If d is a pseudometric rather than a metric, the follow ing modifications must be made. Let (Y,d) be a pseudometric space containing X with Yo E Y-X such that d(yo'x)> 0 for all x E X. Let (Y*,d*) be the metric space formed from (Y,d) in the usual way. Observe that Yo is not identified with any point of X. Let E be the dual of Lip(Y*) and h be defined as above. Then h is still an isometry but not necessarily 1-1, that is h(x) = h(y) iff d(x,y) = O. If xl,···,x are n elements of X such that d(x.,x.)> 0 for i ~ j, the