A constructive approach to minimal projections in Banach spaces

Let X be a Banach space and Y a finite-dimensional subspace of X. Let P be a minimal projection of X onto Y. It is shown (Theorem 1.1) that under certain conditions there exist sequences of finite-dimensional “approximating subspaces” X_m and Y_m of X with corresponding minimal projections P_m: X_m → Y_m, such that lim_m→∞ P_m = P. Moreover, a certain related sequence of projections i_m○P_m○π_m: X → Y has cluster points in the strong operator topology, each of which is a minimal projection of X onto Y. When X = Ca, b] the result reduces to a theorem of 7.]. It is shown (Corollary 1.11) that the hypothesis of Theorem 1.1 holds in many important Banach spaces, including Ca, b], L^Pa, b] and l^P for 1 p < ∞, and c₀, the space of sequences converging to zero in the sup norm.