Nests of subspaces in Banach space and their order types

Arias, Alvaro; Farmer, Jeff

doi:10.1090/S0002-9947-1992-1050084-5

Nests of subspaces in Banach space and their order types
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by Alvaro Arias and Jeff Farmer PDF

Trans. Amer. Math. Soc. 331 (1992), 113-130 Request permission

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 ${L^\infty }[ {0,1} ]$ to a continuous one, a continuous nest in a Banach space having no two elements isomorphic to one another, and a characterization of separable ${\mathcal {L}_p}$-spaces in terms of nests.

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