Vol. 44, No. 2, 1973

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Closed range theorems for convex sets and linear liftings

Tsuyoshi Andô

Vol. 44 (1973), No. 2, 393–410

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.

Mathematical Subject Classification
Primary: 46B05
Received: 13 October 1971
Published: 1 February 1973
Tsuyoshi Andô