Finitely nonreflexive Banach spaces

Abstract

A Banach space X X is said to be finitely nonreflexive if for some n ≥ 1 n \geq 1 the space H n ( X ) {H_n}(X) is reflexive with dim ⁡ H n ( X ) ≥ 1 \dim {H_n}(X) \geq 1 , where H 0 ( X ) = X {H_0}(X) = X , and H n + 1 ( X ) = H n ( X ) ∗ ∗ / H n ( X ) {H_{n + 1}}(X) = {H_n}{(X)^{**}}/{H_n}(X) . A simple construction of (pairwise nonisomorphic) finitely nonreflexive separable Banach spaces which are both nonisomorphic to their Cartesian squares and isomorphic to their biduals is presented.