Convexity Theory and its Applications in Functional Analysis by L. Asimow

By L. Asimow

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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 0 . Thus ±a > 0 and, since Ρ is proper, a - 0 and ρ is a norm. Since I is linearly closed, I = {a : p(a) < 1 } is the unit ball of (E, p).

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||JC, 0 then y - (JC ν 0 ) = y 4- (-JC) A 0 = ( y - j c ) A y 52 DUALITY IN ORDERED BANACH SPACES and (a, y)-(a,x :a vO) = inf {(au y-x) + (a2, y) = ai + a2; au α2>ϋ\ = inf {(a, y ) - (au x) : 0 < ax < a) - (#, y) - s u p {(ai, JC) : 0 < a\< a] and the formula for χ ν 0 is shown.

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.

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