Embeddings of Weakly Compact Sets and *-Paired Banach Spaces

We use properties of Dayʹs norm on c0(κ) to prove that, for every Eberlein compact space K, there exists a separately continuous symmetric mapping d: K×K→R such that we haved(x, y)<d(x, x)+d(y, y)2for any two distinct points x and y of K. As a consequence, we have that every Eberlein compact space K can be embedded as a point-separating set in its own function space C(K) equipped with pointwise (or weak) topology; in the terminology of Arkhangelʹskii, this means that every Eberlein compact space is “self-dual.” We consider whether every Eberlein compact space K can be embedded as a generating set in C(K) (equipped with the weak topology). We show that such an embedding exists for every uniformly Eberlein compact space. We also show that every Eberlein compact space can be embedded as a free generating set in some c0(κ). These results are obtained as special cases of properties of “*-paired Banach spaces,” a notion generalizing the relation of a reflexive Banach space and its dual.