Transitivity of proximinality and norm attaining functionals

Darapaneni Narayana¹ ; T. S. S. R. K. Rao²

¹ Department of Mathematics Indian Institute of Science Bangalore 560012, India
² Stat-Math Unit Indian Statistical Institute R. V. College P.O. Bangalore 560059, India

Colloquium Mathematicum, Tome 104 (2006) no. 1, pp. 1-19

Résumé

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 ${\mathcal K}(\ell _2)$ and $c_0$-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 $c_0$ and show that, for some particular hyperplanes of $c_0$, linearity and orthogonal linearity coincide for the set of norm attaining functionals.

Détail

DOI : 10.4064/cm104-1-1

Keywords: study question set norm attaining functionals banach space linear space property preserved factor reflexive proximinal subspaces widetilde spaces generally taking quotients proximinal subspaces mathcal ell direct sums families reflexive spaces transitivity proximinality factor reflexive subspaces investigate linear structure set norm attaining functionals hyperplanes particular hyperplanes linearity orthogonal linearity coincide set norm attaining functionals