A characterization of Hilbert space

Abstract

A real Banach space E of dimension _3 is an inner product space iff there exists a bounded smooth convex subset of E which is the range of a nonexpansive retraction. De Figueiredo and Karlovitz [3] have shown that if E is a strictly convex real finite-dimensional Banach space and dim E> 3 then there can exist no bounded smooth nonexpansive retract of E unless E is a Hilbert space. (A subset F of E is a nonexpansive retract of E if it is the range of a nonexpansive retraction r: E-F.) This is a consequence of their more general result that if E is reflexive and a convex nonexpansive retract of E has at a boundary point xo a unique supporting hyperplane xo+H then H is the range of a projection of norm 1. As they have pointed out, the latter theorem fails in nonreflexive spaces (the unit ball of C[O, 1] furnishes a counterexample). Nevertheless, their first result is true in general: THEOREM. Suppose E is a real Banach space with dim E> 3. Then E is an inner product space iff there exists a bounded smooth nonexpansive retract of E with nonempty interior. We separate out of the proof of the theorem a lemma, valid in all real Banach spaces: LEMMA. Suppose F is a bounded smooth closed convex subset of a real Banach space E and F has nonempty interior. Then given disjoint bounded closed convex sets M and K in E with K compact, there exist p E E and 2>0 such that Kcp+)LF and (p+ 2F) rnM= 0. PROOF OF LEMMA. Clearly the hypotheses and conclusions of the lemma are invariant if K and M are translated by the same vector; thus without loss of generality we may assume 0 E K. Similarly, we may also assume 0 E int F. Since K is compact and M is closed, a basic separation theorem for convex sets assures the existence of a closed hyperplane H which strictly separates M and K; that is, there exist we E*, c e R' Received by the editors June 26, 1972 and, in revised form, August 21, 1973. AMS (MOS) subject classifications (1970). Primary 46C05.