By L. Asimow
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5. T H E O R E M . Let u be an Archimedean order unit for (Ε, P) and let I and ρ be as above. Then ρ is a norm on Ε with unit ball I in which Ε is a normed order unit space with order unit u and closed convex positive cone P. Proof. The set I is symmetric and convex. 4(ii) and (iv) shows 0 e core I and I is linearly closed. Hence ρ (I) is a semi-norm. If ρ (a) = 0 then λα e I for all Λ ^ 0 and hence -u
Since Ε is a-generated u> has a unique extension to P - P = Ε such that if c = a-b; a,beP and ||a|| + ||6||
If Ε is an ordered norm space then the following are equivalent: (i) Ε is (a, n)-additive; (ii) E* is (a, n)-directed. 9. COROLLARY. The following are equivalent: (i) Ε is a base norm space ; (ii) jB* is an order unit space. If these conditions hold and Β is the base for Ρ and u the order unit for E* then B={aeP:(a, w) = l } . B it is easy to check that the order unit space E* In the case when Ε = A(K)* coincides with the space A (K) of all bounded real-valued affine functions on Κ with the supremum norm.