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

By L. Asimow

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Example text

The following are equivalent: BANACH LATTICES AND THE RIESZ INTERPOLATION PROPERTY (i) (Ε, P) is a lattice ; (ii) a,bePimplies there is a unique a AbeP. Proof. Assume (ii) holds. Then given any c,deE b o t h c + e, d + eeP. Then c Λ 49 there is an e e Ε such that d = (c + e) Λ (d + e) - e shows c Λ d exists for any pair c, d. Then Cs/d = - [ ( - c ) Λ (-+0 ifterc c^a~; α + + α~ = |α| = α ν α ~ ; PROPOSITION.

Then for each 0 < y < 1 there is a support pair (b, y ) (||y || = 1) with \\b-a\\

Since the sets Ει ± P * are already w*-closed we can take the polar of the intersection and get the equivalent statement (iii) Ει c α co (Pi u - P i ) 34 DUALITY IN O R D E R E D BANACH SPACES But then Β = co (Pi u - P i ) is a regular pre-gauge set and hence (iii) is equivalent to Εχ c a' co (Pi u - P i ) for all α' > a, which is precisely (ii). 5 is the best conclusion. 6. EXAMPLE. Let c 0 denote the Banach space of sequences χ = {xn)™=\ such that lim n-oo xn~0 with ||*|| = max{|*„|: η = 1 , 2 , .

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