Projections in a normed linear space and a generalization of the pseudo-inverse

Abstract

The concept of a in a finite-dimensional real or complex normed linear space H (the function PM which carries every element into the closest element of a given subspace M) is set forth and examined. If dim M = dim H - 1, then PM is linear. If PN is linear for all k-dimensional subspaces N, where 1 ≤ k The projective bound Q, defined to be the supremum of the operator norm of PM for all subspaces, is in the range 1 ≤ Q < 2, and these limits are the best possible. For norms with Q = 1, PM is always linear, and a characterization of those norms is given. If H also has an inner product (defined independently of the norm), so that a dual norm can be defined, then when PM is linear its adjoint PMH is the projection on (kernel PM)⊥ by the dual norm. The projective bounds of a norm and its dual are equal. The notion of a pseudo-inverse F+ of a linear transformation F is extended to non-Euclidean norms. The distance from F to the set of linear transformations G of lower rank (in the sense of the operator norm ∥F - G∥) is c/∥F+∥, where c = 1 if the range of F fills its space, and 1 ≤ c In all results, the real and complex cases are handled in a completely parallel fashion.