Permanence properties of normal structure

Thomas Landes

Vol. 110 (1984), No. 1, 125–143

DOI: 10.2140/pjm.1984.110.125

Abstract

A new characterization of normal structure is given, which allows to prove permanence properties of normal structure such as preservation under finite direct-sum-operations — e.g., the l_p^N-direct sums, 1 < p ≤∞ - as well as under certain infinite direct-sum-operations — e.g., the l_p-direct sums, 1 < p < ∞.

Furthermore, it is shown that a normed space has isonormal structure — i.e., it is isomorphic to a normally structured space — if and only if it can be mapped by a continuous linear one-to-one operator into some normally structured space.

Finally, some problems are discussed, such as preservation of normal structure under the l₁²-direct-sum-operation. To solve the latter at least partially, a sum-property is introduced which implies normal structure. This sum-property is implied by all known sufficient conditions for normal structure, and it is preserved under all finite direct-sum-operations.

Mathematical Subject Classification 2000

Primary: 46B20

Secondary: 47H10

Milestones

Received: 8 February 1982

Revised: 10 March 1982

Published: 1 January 1984