Local Numerical Invariants of Banach Spaces

Abstract:

 In 1932, Banach devoted Chapter 12 of his book “Theory of Linear Operations” to determining the values of pp and qq for which the Banach space Lq[0,1]Lq[0,1] is (linearly) isomorphic to a subspace of Lp[0,1]Lp[0,1]. This question has since been answered in many different ways. A modern approach to answering this question is by means of local numerical invariants of Banach spaces. The reason that such an approach is desirable is that a certain rigidity theorem of Ribe indicates that such local numerical invariants should also have metric space analogues. In this talk we discuss the basic theory of the classical local numerical invariants type and cotype, as well as the more recent numerical invariant known as XpXp type, the construction of which is due to Naor and Schechtman (2016).