On some new metric characterisations of Hilbert spaces

TL;DR

This paper surveys recent Lipschitz-based characterizations of Hilbert spaces, focusing on isomorphic conditions involving equivalent norms derived from inner products, contrasting with traditional linear theory approaches.

Contribution

It introduces Lipschitz analogues of classical linear characterizations of Hilbert spaces, emphasizing isomorphic conditions with equivalent inner product-induced norms.

Findings

Lipschitz analogues of linear Hilbert space characterizations

Focus on isomorphic conditions with equivalent inner product norms

Survey of recent results in Lipschitz category theory

Abstract

In the literature surrounding the theory of Banach spaces, considerable effort has been invested in exploring the conditions on a Banach space X that characterise X as being an inner product space or as a linearly isomorphic copy of a Hilbert space. On the other hand, a different theory emerges when the class of Banach spaces is looked upon as a Lipschitz category where Lipschitz maps are used as morphisms in the new category in place of the familiar bounded linear maps in the linear theory. This paper provides a short survey of recent results involving the appropriate Lipschitz analogues of certain well known results from the linear theory characterizing Hilbert spaces. Whereas isometric description of Hilbert spaces has all along been a popular theme in this line of investigations, we shall concentrate mainly on isomorphic characterisations which entail the existence of an equivalent norm on the underlying space arising from an inner product.