Bases in Banach Spaces II

Singer, Ivan

doi:10.1007/978-3-642-67844-8

Cited by 340 publications

(108 citation statements)

References 0 publications

Supporting

Mentioning

104

Contrasting

Unclassified

Order By: Relevance

“…In other words T ξn 's converge to T ξ in the strong operator topology and so, the inductive assumption implies that T ξ is in the closure of the linear span of operators of the form ρ α ξ (0) [P α ξ (1) − P α ξ (0) ] for ξ < η which completes the proof of the claim. Note that we actually proved that it is the limit of a transfinite sequence of these operators in the sense of [15].…”

Section: Proof Of the Claim 1 It Is Clear Thatmentioning

confidence: 72%

Projections in Weakly Compactly Generated Banach Spaces and Chang's Conjecture

Koszmider¹

2005

Journal of Applied Analysis

View full text Add to dashboard Cite

Abstract. Classical results on weakly compactly generated (WCG) Banach spaces imply the existence of projectional resolutions of identity (PRI) and the existence of many projections on separable subspaces (SCP). We address the questions if these can be the only projections in a nonseparable WCG space, in the sense that there is a PRI (Pα : ω ≤ α ≤ λ) such that any projection is the sum of an operator in the closure of the linear span of countably many Pα's (in the strong operator topology) and a separable range operator. Wark's modification of Shelah's and Steprāns' construction provides an unconditional example for λ = ω1. We note that it is impossible for λ > ω2. . The main result of the paper is that for λ = ω2, the second uncountable cardinal, the question is logically undecidable and depends on additional axioms deciding the combinatorics on ω2; for example Chang's conjecture implies that there are other projections than the projections mentioned above. The full strength results concern all linear operators not just the projections.

show abstract

Section: Proof Of the Claim 1 It Is Clear Thatmentioning

confidence: 72%

Projections in Weakly Compactly Generated Banach Spaces and Chang's Conjecture

Koszmider¹

2005

Journal of Applied Analysis

View full text Add to dashboard Cite

show abstract

“…If (x n , f * n ) is a bibasic sequence which satisfies (vii), then, by Proposition 1.14 on p. 91 of [14], [x n ] ⊥ +[f unconditionality (or some other property) has been omitted from the description of the basic sequence in the problem.…”

Section: Theorem 3 If K Is a Nonrelatively Weakly Compact Dunford-pementioning

confidence: 99%

“…An application of Lemma 1.3 of [14] or Proposition 3 of [4] ensures that (x * n ) ∼ (f * n ), and (vii) implies (ii). Now suppose that (iv) holds.…”

mentioning

confidence: 99%

Dunford–Pettis sets

Lewis

2001

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Bibasic sequences are used to study relative weak compactness and relative norm compactness of Dunford-Pettis sets.A Banach space X has the Dunford-Pettis property provided that every weakly compact operator with domain X and range an arbitrary Banach space Y maps weakly compact sets in X into norm compact sets in Y . Localizing this notion, a bounded subset A of X is said to be a Dunford-Pettis subset of X if T (A) is relatively norm compact in Y whenever T : X → Y is a weakly compact operator. Consequently, a Banach space X has the Dunford-Pettis property if and only if each of its weakly compact sets is a Dunford-Pettis set. The survey article by Diestel [5] is an excellent source of information about classical results in Banach spaces which relate to the Dunford-Pettis property.Kevin Andrews utilized Dunford-Pettis sets in a study of the Bochner integral in E. Bator showed in [2] that a dual space has the weak Radon-Nikodym property if and only if each Dunford-Pettis subset of X is relatively compact. In addition to reproducing Bator's result, Emmanuele [8] established several other structure properties for Banach spaces in which all Dunford-Pettis sets are relatively compact. Since every bounded subset of a Banach space X whose dual space X * has the Schur property is a Dunford-Pettis subset of X, it is clear that there are Dunford-Pettis sets which are not relatively weakly compact. However, we note that Odell [13, p. 377] showed that every sequence in a Dunford-Pettis set has a weakly Cauchy subsequence. In this paper we study Dunford-Pettis sets which fail to be relatively norm or weakly compact.

show abstract

“…])• First, we need an auxiliary lemma. But let us remember that a system {x,} in a Banach space X is called an M-basis of the space X if {x,} is the minimal complete system with total biorthogonal system [7].…”

Section: On Supportless Convex Sets In Incomplete Normed Spaces V Pmentioning

confidence: 99%