Nests of subspaces in Banach space and their order types

Abstract

This paper addresses some questions which arise naturally in the theory of nests of subspaces in Banach space. The order topology on the index set of a nest is discussed, as well as the method of spatial indexing by a vector; sufficient geometric conditions for the existence of such a vector are found. It is then shown that a continuous nest exists in any Banach space. Applications and examples follow; in particular, an extension of the Volterra nest in L ∞ [ 0 , 1 ] {L^\infty }[ {0,1} ] to a continuous one, a continuous nest in a Banach space having no two elements isomorphic to one another, and a characterization of separable L p {\mathcal {L}_p} -spaces in terms of nests.