Finite co-dimensional Banach spaces and some bounded recovery problems

Abstract

In this paper we study the projections and some recovery problem of a finite co-dimensional Banach spaces in terms of the projection of their complementations, more precisely we study the following problems: (1) If Y is a finite co-dimensional subspace of a Banach space X and Z is its complementation, is for every projection P0 from X onto Z and every ε > 0 there a projection P from X onto Y satisfying ∥P∥≤ 1 + (1 + ε)∥P0∥? (2) If X is a Banach space, x ∈ X, Y is an n-co-dimensional subspace of X and ({fi,xi} i=1n) is the Auerbach system of the complementation Z of Y in X, is there an element y ∈ Y satisfying the following two conditions (i) f̂i(y) = f̂i(x)∀i ∈ {1,2,...,n}, where f̂i is the Hahn-Banach extension of fi on X, (ii) ∥y∥≤M∥x∥ for some constant M? And we study the restrictions placed on the constant M as a function of X and Y. © 2003 Elsevier Inc. All rights reserved.