Optimal and one-complemented subspaces

On intersections of ranges of projections of norm one in Banach spaces

“…From Lindenstrauss' characterization of L 1 -preduals in terms of every collection of 4 pairwise intersecting balls having non-empty intersection (see [13], page 58), it follows that if Y ⊂ X is a central subspace and X is an L 1 -predual, then Y is an L 1 -predual. This notion is weaker than that of an existence set considered in [12], where the inequalities in the definition should hold for all y ∈ Y . Thus the following result extends Theorem 2.3 in [12].…”

On ideals and generalized centers of finite sets in Banach spaces

“…We will be using the notion of an existence set and a result due to Lewicki and Trombetta [3] in the proof of the next result. Since Y is an ideal in Z , let P : Z * * → Y ⊥⊥ be a projection of norm 1.…”

“…In [2] Lindenstrauss gave an example of a space Y isometric to a space of continuous functions on a compact set, and a Banach space Z such that Y ⊂ Z is of codimension 2 and Y is not the range of a projection of norm 1 in Z , but for every z ̸ ∈ Y , Y is the range of a projection of norm 1 in span{z, Y }. Subspaces that satisfy the hypothesis in the Lindenstrauss problem were (in an equivalent formulation) called existence sets in [3], where the authors give a positive solution to this problem for subspaces of several function spaces.…”

“…We first show that Y is the range of a projection of norm 1 in span{x, Y }. It would then follow from Lemma 0.1 in[3] that Y is an existence set. It then follows from Theorem 2.4 in[3] that Y is the range of a projection of norm 1 in ℓ 1 .Let x ̸ ∈ Y and let Z = span{x, Y }.…”

“…It would then follow from Lemma 0.1 in[3] that Y is an existence set. It then follows from Theorem 2.4 in[3] that Y is the range of a projection of norm 1 in ℓ 1 .Let x ̸ ∈ Y and let Z = span{x, Y }. It follows from the proof of Lemma 5.2 in[1] that if ∩ y∈Y B Y (y, ∥x − y∥) ̸ = ∅, then there is a projection of norm 1 from Z onto Y .…”