ABB Theorems: Results and Limitations in Infinite Dimensions
在ℓ²空间中构造了一个弱紧凸集，其具有孤立的最大元且不能被严格正泛函支撑，表明Arrow-Barankin-Blackwell定理在无限维中不成立，并指出锥有界基是定理成立的关键条件。
Abstract We construct a weakly compact convex subset of $$\ell ^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> with nonempty interior that has an isolated maximal element, with respect to the lattice order $$\ell _{+}^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℓ</mml:mi> <mml:mrow> <mml:mo>+</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> . Moreover, the maximal point cannot be supported by any strictly positive functional, which shows that the Arrow-Barankin-Blackwell theorem fails. This example discloses the pertinence of the assumption that the cone has a bounded base for the validity of the result in infinite dimensions. Under this latter assumption, the equivalence of the notions of strict maximality and maximality is established.