Let M be a closed subspace
of a Banach space E such that its annihilator M⊥ is the range of a projection P.
Given a closed convex subset S containing 0, the first problem of this paper is to find
a condition for τ(S) to be closed where τ is the canonical map from E to E∕M.
Closure is guaranteed if S is splittable in the sense that the polar S0 coincides with
the norm-closed convex hull of P(S0) ∪ Q(S0), where Q = 1 − P. The second
problem is to give a condition for existence of a linear map φ, called a linear
lifting, from E∕M to E such that τ ∘ φ = 1 and φ ∘ τ(S) ⊆ S. A linear lifting
exists if and only if M is the kernel of a projection making S invariant. Of
special interest is the case where lS is a ball or a cone. When the unit ball is
splittable, existence of a linear lifting of norm one is guaranteed under suitable
conditions on E∕M, which are satisfied by separable Lp and C(X) on compact
metrizable X. If further E is an ordered Banach space, and if both P and Q
are positive, M is shown to be the kernel of a positive projection of norm
one.
