On<i>k</i>-shrinking and<i>k</i>-boundedly complete basic sequences and quasi-reflexive spaces

Sternbach, Leonard

doi:10.2140/pjm.1971.37.817

Cited by 2 publications

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“…And continue this until X/[Xk] is reflexive, Remark. This includes both results of Sternbach [14] and a result of McWilliams [12]. Corollary 14 (Clark [2]).…”

Section: Let Bi=en(o and Q~: [Bi]~[e] Be The Injection And Hence Q~*:...mentioning

confidence: 91%

Somewhat quasireflexive Banach spaces

Bellenot

1984

Ark. Mat.

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The question of "What kind of subspaces must a nonreflexive Banach space X have?" has received a lot of attention. Pelczynski [23] (in 1962) has given the most general answer to date: X contains a basic sequence which is not shrinking (and hence spanning a nonreflexive space). For special cases more is known. Johnson and Rosenthal [8] have shown that X and X* contain reflexive subspaces if X** is separable. (This was extended to the case when X**/X is separable by Clark [2].) In another direction, Davis and Johnson [5] have shown that if X**/X is infinite dimensional then X contains a basic sequence that spans a nonquasireflexive subspace. Perhaps the main reason for this interest are the following two long open questions:(1) Does each Banach space contain an unconditional basic sequence? (2) Does each Banach space contain a subspace isomorphic to c0,/1 or to a reflexive space?Indeed, James [6] has shown that a positive answer to (1) implies a positive answer to (2). And clearly these results are partial answers to (2). On the other hand, consider the collection of spaces to which the special cases apply. James [7], Lindenstrauss [9], Davis, Figiel, Johnson and Pelczynski [4] and the author [1] show how to construct an X so that X**/X is a pregiven Z (with restrictions on Z). All these constructions depend on reflexivity or quasireflexivity in a strong way and the constructed X has lots of quasireflexive subspaces. This paper attempts to unite these results. It is shown that if X**/X is separable then each element of X**/X is "reachable" by an order one quasireflexive subspace ZcX, so that Z has a shrinking basis (Theorem 8). If X** is separable, both X and X* have subspaces and quotients which are order one quasireflexive with bases (Theorem 9). And if X* is separable then X has a nonreflexive quotient

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“…And continue this until X/[Xk] is reflexive, Remark. This includes both results of Sternbach [14] and a result of McWilliams [12]. Corollary 14 (Clark [2]).…”

Section: Let Bi=en(o and Q~: [Bi]~[e] Be The Injection And Hence Q~*:...mentioning

confidence: 91%