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2 Colloquium Mathematicum

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Transitivity of proximinality and norm attaining functionals

100%

Narayana D. , Rao T.

nr 1

1-19

We study the question of when the set of norm attaining functionals on a Banach space is a linear space. We show that this property is preserved by factor reflexive proximinal subspaces in $\widetilde{R(1)}$ spaces and generally by taking quotients by proximinal subspaces. We show, for 𝒦 (ℓ₂) and c₀-direct sums of families of reflexive spaces, the transitivity of proximinality for factor reflexive subspaces. We also investigate the linear structure of the set of norm attaining functionals on hyperplanes of c₀ and show that, for some particular hyperplanes of c₀, linearity and orthogonal linearity coincide for the set of norm attaining functionals.

Ball remotal subspaces of Banach spaces

80%

Bandyopadhyay P. , Lin B.-L. , Rao T.

Colloquium Mathematicum

2009

tom 114

nr 1

119-133

We study Banach spaces X with subspaces Y whose unit ball is densely remotal in X. We show that for several classes of Banach spaces, the unit ball of the space of compact operators is densely remotal in the space of bounded operators. We also show that for several classical Banach spaces, the unit ball is densely remotal in the duals of higher even order. We show that for a separable remotal set E ⊆ X, the set of Bochner integrable functions with values in E is a remotal set in L¹(μ,X).