# ABB Theorems: Results and Limitations in Infinite Dimensions

## TL;DR

This paper explores mathematical properties of infinite-dimensional spaces and shows when a certain theorem fails.

## Contribution

The paper constructs a counterexample showing the Arrow-Barankin-Blackwell theorem fails in infinite dimensions.

## Key findings

- A weakly compact convex set in ℓ² has an isolated maximal element under lattice order.
- The maximal point cannot be supported by any strictly positive functional.
- The theorem's validity requires the cone to have a bounded base in infinite dimensions.

## Abstract

We construct a weakly compact convex subset of \documentclass[12pt]{minimal}
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\begin{document}$$\ell ^{2}$$\end{document}ℓ2 with nonempty interior that has an isolated maximal element, with respect to the lattice order \documentclass[12pt]{minimal}
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\begin{document}$$\ell _{+}^{2}$$\end{document}ℓ+2. 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.