A constructive approach to minimal projections in Banach spaces

Abstract

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 = C[ a, b] the result reduces to a theorem of Cheney and Morris (“The Numerical Determination of Projection Constants,” Report No. 75, Center for Numerical Analysis, The University of Texas at Austin, 1973). It is shown (Corollary 1.11) that the hypothesis of Theorem 1.1 holds in many important Banach spaces, including C[ a, b], L P [ a, b] and l P for 1 ⩽ p < ∞, and c 0, the space of sequences converging to zero in the sup norm.