Directionally Euclidean structures of Banach spaces

Jarno Talponen¹

¹ Institute of Mathematics Aalto University P.O. Box 11100 FI-00076 Aalto, Finland

Studia Mathematica, Tome 202 (2011) no. 2, pp. 191-203

Résumé

We study Banach spaces with directionally asymptotically controlled ellipsoid-approximations of the unit ball in finite-dimensional sections. Here these ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding finite-dimensional subspace. The directional control here means that we evaluate the ellipsoids by means of a given functional of the dual space. The term `asymptotical' refers to the fact that we take `$\limsup$' over finite-dimensional subspaces. This leads to isomorphic and isometric characterizations of Hilbert spaces. An application involving Mazur's rotation problem is given. We also discuss the stability of the family of ellipsoids as the dimension and geometry vary. The methods exploit ultrafilter techniques and we also apply them in conjunction with finite Auerbach bases to study the convexity properties of the duality mappings.

Détail

DOI : 10.4064/sm202-2-5

Keywords: study banach spaces directionally asymptotically controlled ellipsoid approximations unit ball finite dimensional sections here these ellipsoids unique minimum volume ellipsoids which contain unit ball corresponding finite dimensional subspace directional control here means evaluate ellipsoids means given functional dual space term asymptotical refers limsup finite dimensional subspaces leads isomorphic isometric characterizations hilbert spaces application involving mazurs rotation problem given discuss stability family ellipsoids dimension geometry vary methods exploit ultrafilter techniques apply conjunction finite auerbach bases study convexity properties duality mappings