JOURNAL OF APPROXIMATION THEORY 41, 367-381 (1984) Minimal Projections in Tensor-Product Spaces C. FRANCHETTI * Istituto di Matematica Applicata, Facoltd di Ingegneria, Vniversitd degli Studi di Firenre, SO139 Firenre, Italia AND E. W. CHENEY Department of Mathematics, University of Texas, Austin, Texas 78712, U.S.A. Communicated by Oved Shisha Received July 3, 1983; revised October 14, 1983 1. INTRODUCTION A projection of a Banach space X onto a subspace V is a bounded linear map P: X- V such that P2 = P. (The arrow with two heads denotes a surjective map.) For many applications, a projection with nearly minimal norm is sought. The greatest lower bound for lIPI is the relative projection constant of V in X: A( V, X) = inf{lI PII: P E .i/‘(X, V), P(X) = V, P2 = P). The absolute projection constant of a Banach space Y is defined by l(Y) = sLlp(/qY, Z): z 3 Y}. These numbers may be infinite. Our interest here is in the projection constants of subspaces of tensor- product spaces. For example, if G c X and H c Y (all Banach spaces), how is ;1(G @ H, X @ Y) related to I1(G, X) and A(H, Y)? This problem does not become properly posed until the topology of X@ Y has been specified. It is convenient to assume that a reasonable norm a has been defined on the ‘t During the preparation of this paper Dr. Franchetti was a visiting professor at the University of Texas. 367 0021.9045/84 $3.00 Copyright ‘? 1984 by Academx Press, Inc. All rights of reproduction m any form reserved.