ABB Theorems: Results and Limitations in Infinite Dimensions.

Area of Science:

• Functional Analysis
• Convex Analysis
• Set Theory

Background:

• The Arrow-Barankin-Blackwell theorem is crucial for understanding maximal elements in ordered spaces.
• Infinite-dimensional spaces present unique challenges for classical theorems.
• The properties of convex sets and cones are fundamental in optimization and analysis.

Purpose of the Study:

• To construct a specific type of convex set in ℓ² space.
• To investigate the conditions under which maximal elements are isolated.
• To test the applicability of the Arrow-Barankin-Blackwell theorem in infinite dimensions.

Main Methods:

• Construction of a weakly compact convex subset of ℓ².
• Utilizing lattice ordering (ℓ²₊) to define maximality.
• Analysis of supporting functionals for elements within the set.

Main Results:

• A weakly compact convex subset of ℓ² with a nonempty interior and an isolated maximal element was successfully constructed.
• The constructed maximal element could not be supported by any strictly positive functional.
• This finding demonstrates a failure of the Arrow-Barankin-Blackwell theorem in this specific infinite-dimensional context.

Conclusions:

• The example highlights the importance of the 'bounded base' assumption for cones in infinite-dimensional spaces.
• The Arrow-Barankin-Blackwell theorem's validity is contingent on specific geometric properties of the underlying cone.
• Under the assumption of a bounded base, maximality and strict maximality are shown to be equivalent concepts.