Banach spaces with the extension property

Abstract

Recently, in these Transactions, Nachbin [N] and, independently, Goodner [G] have shown that if B has the extension property and if its unit sphere has an extreme point, then B is equivalent to a function space of this sort; both authors have also proved that such a function space has the extension property. The above theorem simply omits the extreme point hypothesis, and so establishes the equivalence. My original proof, of which the proof given here is a distillate, depends on an idea of Jerison [j]. Briefly, letting X be the weak* closure of the set of extreme points of the unit sphere of the adjoint B*, B can be shown equivalent to the space of all weak* continuous real functions / on X such that/(x) = —f( — x), and then properties of X are deduced which imply the theorem. The same idea occurs implicitly in the proof below. Note. Goodner asks [G, p. 107] if every Banach space having the extension property is equivalent to the conjugate of an abstract (L)-space. It is known (this is not my contribution) that the Birkhoff-Ulam example ([B, p. 186] or [HT, p. 490]) answers this question in the negative, the pertinent Banach space being the bounded Borel functions on [0, 1 ] modulo those functions vanishing except on a set of the first category, with ||/|| = inf {K: \f(x) | g K save on a set of first category}. 1. Preliminary definitions and remarks. A point x is an extreme point of a convex subset K of a real linear space if x is not an interior point of any line segment contained in K (i.e., if x=ty + (l — t)z, 0<t<l, y£K, and z(E.K, then x=y=z). A set I is a support of K if L is a convex, nonvoid subset of K such that each line segment contained in K which has an interior point in L is contained in L. If x is an extreme point of L and L is a sup-