The Kreps-Yan theorem for L∞

Let 〈X ,Y〉 be a pair of Banach spaces in separating duality [18, Chapter IV]. A convex set M ⊂ X is called cone if λx ∈M for any x ∈M, λ ≥ 0. A cone M is called pointed if M∩ (−M)= {0}. Suppose that X is endowed with a locally convex topology τ, which is always assumed to be compatible with the duality 〈X ,Y〉, and K ⊂ X is a τ-closed pointed cone. An element ξ ∈ Y is called strictly positive if 〈x,ξ〉 > 0 for all x ∈ K\{0}. An element ξ is called nonnegative if 〈x,ξ〉 ≥ 0 for all x ∈ K . We only consider cones K such that the set of strictly positive functionals is nonempty. Following [10], we say that the Kreps-Yan theorem is valid for the ordered space (X ,K) with the topology τ if for any τ-closed convex cone C, containing −K , the condition C∩ K = {0} implies the existence of a strictly positive element ξ ∈ Y such that its restriction on C is nonpositive: 〈x,ξ〉 ≤ 0, x ∈ C. We also refer to [10] for the comments on the papers of Kreps [12] and Yan [20]. If the above statement is true for any τ-closed pointed cone K ⊂ X , we say that the Kreps-Yan theorem is valid for the space (X ,τ). It should be mentioned that in this terminology the Kreps-Yan theorem may be valid for (X ,τ) even if there exists a τ-closed pointed cone such that the set of strictly positive functionals is empty. Recall that a space (X ,τ′) is said to be Lindelöf, or have the Lindelöf property, if every open cover of X has a countable subcover [11]. As usual, we denote the weak topology by σ(X ,Y). The next theorem is, in fact, a partial case of [10, Theorem 3.1].