By Frederick S. Woods
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Proof. i. must be normal. Also the interior of X+ is non-empty so that X has an order unit. 1 0 and the equivalence of all norms on finite dimensional spaces. The convex hull of these points is an n-simplex, S. If P1 is the (closed) cone with base 51 then clearly P1C X 4-, and Pi- P1 = X. To find P2 first find P * C- X + closed, generating and , 1 4i * 7:- X + X inducing a lattice ordering. Now let P 2 = P1 ** ( identifying X with x** ). 1 P2 induces a lattice ordering on X. Clearly P 2 is closed and generating so the result is complete.
1 may be repeated. 12. 4. K(X,Y) is positvely generated wtonever X. is base-normed, if and only if each relatively compact subset of Y is order-bounded. 53 Proof. If X is base-nonmed, with base B, let T E K(X,Y). It follows that T(B) is a relatively compact subset of Y. Hence so also is co( CO1 k-1 T(B)). Let e be an upper bound for this latter set. : 0 f 113 1. 12 and define S e K(X,Y) by S : xi--+f(x)e. Clearly if b E S o Sb,Tb, O. But if xe X÷ , then x= X b for some X >,0, and b EB. Thus we see that S T 0 0.
A)(b) : As Y is assumed to be positively generated, it will suffice to show that Y is conditionally canplete. Let g be a non-zero bounded positive linear functional on X , with 44 g(x) :: 1 1 for x 0 E 4 0 and II :x 0 t l. Let (s i) i el be a family II ' in Y, bounded above by M. Define Si : g(x)si, with M defined similarly. If x), 0 , then (X S i )x - s i )g(x) so that RE S i . Let T be the supremum of (sdi which exists el 0, in L(X,Y), by assumption. As T S i, we have Tx. >, Six, On the other band, if t defining T' si.
A course in mathematics,: For students of engineering and applied science, by Frederick S. Woods