Finite co-dimensional Banach spaces and some bounded recovery problems

摘要

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.