There is no bilinear Bishop-Phelps theorem

Acosta, Marı́a D.; Aguirre, Francisco Javier; Payá, Rafael

doi:10.1007/bf02761104

Cited by 50 publications

(45 citation statements)

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“…In 1998, Acosta, Aguirre and Payá [3] found the first negative example. In 1997, Alaminos, Choi, Kim and Payá [8] showed that norm attaining bilinear forms on C 0 (L) spaces are dense in the space of all bounded bilinear forms.…”

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confidence: 99%

Bishop–Phelps–Bollobás property for bilinear forms on spaces of continuous functions

2015

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Abstract. It is shown that the Bishop-Phelps-Bollobás theorem holds for bilinear forms on the complex C 0 (L 1 ) × C 0 (L 2 ) for arbitrary locally compact topological Hausdorff spaces L 1 and L 2 .All along the paper, we will use the following usual notation. Let B X and S X denote, respectively, the closed unit ball and the unit sphere of a Banach space X, and let X * denotes the (topological) dual space of X. We write L(X, Y ) to denote the space of all bounded linear operators from a Banach space X into a Banach space Y and B(X × Y ) to denote the space of all bounded bilinear forms defined on X × Y . We say that T ∈ L(X, Y ) (respectively B ∈ B(X × Y )) attains its norm if there is x ∈ S X such that T x = T (respectively, there are x ∈ S X and y ∈ S Y such that |B(x, y)| = B ).The classical Bishop-Phelps theorem [14] states that the set of norm attaining linear functionals are dense in the topological dual of an arbitrary Banach space. Bollobás Let X be a Banach space. If x ∈ B X and x * ∈ S B * X satisfy Re x * (x) > 1 − ε 2 /2. Then there exist y ∈ S X and y * ∈ S X * such that y * (y) = 1, x * − y * < ε and x − y < ε. there exist x 1 ∈ S X and S ∈ L(X, Y ) such that S(x 1 ) = S = 1, x 0 − x 1 < ε and S − B < ε. In this case, it also said that the Bishop-Phelps-Bollobás theorem holds for L(X, Y ).It is clear that the Bishop-Phelps-Bollobás property for a pair (X, Y ) implies that norm attaining operators are dense in L(X, Y ), being false the converse: there is a (reflexive) space Y such that the

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confidence: 99%

Bishop–Phelps–Bollobás property for bilinear forms on spaces of continuous functions

2015

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“…Unlike the linear case, the answer to this question is negative, an example of a Banach space X such that Ꮽᏸ 2 (X) is not dense in ᏸ 2 (X) was recently exhibited by Acosta, Aguirre, and Payá [1].…”

Section: Introduction a Classical Results Of Bishop And Phelpsmentioning

confidence: 99%

Norm attaining bilinear forms on L₁(μ)

Saleh

2000

International Journal of Mathematics and Mathematical Sciences

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“…In general, however, part a) of the three questions has a negative answer. In [2], Acosta, Aguirre, and Payá showed that the answer to the first part of questions (i) and (ii) is negative for X = G, the Gowers space. An independent example was given by Choi [7] for…”

Section: §1 Introductionmentioning

confidence: 99%

“…Moreover, Jiménez and Payá showed that the set N A(P( n X)) of norm attaining n-homogeneous polynomials on a Banach space X is not norm dense in the Banach space P( n X) of all nhomogeneous polynomials on X ([12, Theorem 3.2], see also [2]). The situation for part b) of these questions is quite different.…”

Section: §1 Introductionmentioning

confidence: 99%