LINEARITY OF THE NEAREST POINT CROSS SECTION OPERATORS ON HOLOMORPHIC FUNCTION SPACES By J. A. CIMA AND M.I. STESSIN O. Introduction Let X be a closed subspace of a Banach space Y. A map M (which in general is nonlinear) from Y/X into Y is called the nearest point cross section ifM(w) E w and Ilwlb'/x = IIM(w)[Ir for every w E Y/X. In general, the nearest cross section need not exist; but there are interesting situations where it does exist. One such situation is the case of the classical Hardy space Hp of the circle, p > 1. When p = 1 the nearest cross section provides an efficacious tool in characterizing weakly compact subsets of L 1/Hg and weakly compact subsets of the dual of the disk algebra (see Pelczynski [4]). Results of this type have also been used by J. Bourgain [1] in his proof that H a of the unit disk has the Dunford-Pettis property. In this paper we consider for Y the Bergman space LP(f~) (1 < p < oc) where f~ is a bounded domain in C n and the vectors are holomorphic functions (in f~) which belong to L p with respect to Lebesgue measure on the domain. In the case of a bounded strictly pseudoconvex domain f~, similar results are valid for Hardy. spaces Hp(f~). Since all proofs are literally the same, we focus on the Bergman case. For a given subset A of f2, we consider the subspace HA = {f E LP(f2) :f(w) = 0 for w E A}. The nearest P LP(f~) is a minimizing point cross section mapping M~ for the pair A/'A C operator given as follows. Fix F in LP(ft) and let .A/[F ~--- {f E LP(a) :flA = FIA} . Write (1) m(F) = inf{l]fllp :f E .MF}. There are basic facts that are needed. First, if we denote by the same letter F the coset F + HA of the quotient space LP(f~)/.MA, then MP(F) is a function in F and 221 JOURNAl. D'ANALYSE MATH~MATIQUE, Vol. 65 (1995)