Continuous metric projections

Abstract

An example is given of a reflexive, rotund Banach space whose dual space is not Frechet differentiable such that every metric projection onto closed subspaces is norm continuous. This example shows that several published conjectures on necessary and sufficient conditions for a reflexive, rotund Banach space to have norm continuous metric projections onto all closed subspaces are incorrect. Introduction. If X is a reflexivee, rotund Banach space, every closed subspace of X is a unique best approximation subspace, called a Chebyshev subspace. If M is a closed subspace of X, the metric projection PM associated with M is defined via infmeM lix ml| = lix Pm(x)i|. An open question in best approximation theory is to find necessary and sufficient conditions on a reflexive, rotund Banach space to insure that every closed subspace has a norm continuous metric projection associated with it. In [51, the conjecture was that every reflexive and rotund Banach space had this property. In [41, this was later amended to be that Banach spaces with a Frechet smooth dual space had this property. In [41, it was also conjectured that if the canonical duality map from X* to X, restricted to M' = Xx* e X*ix*(m) = 0, Vm c MI, was norm continuous, then PM was norm continuous. It was verified for subspaces M of finite codimension in [4]. In [1], the study of bounded compactness led to the continuity of PM for those M such that ker PM = LX x X I PM(x) = 01 was boundedly compact. Further, if the codimension of M was finite, then the bounded compactness of ker P. was a necessary and sufficient condition that PM be norm continuous. In this paper, we show that there exists a reflexive, rotund Banach space whose dual is not Fre'chet smooth and yet every metric projection is norm continuous. This is accomplished by closely examining the space X exhibited by Klee in [61, where X is a renorm of 12, which is reflexive, Gateaux smooth at all points of the unit ball except t80, 80}. By well-known duality relationships, X* is a reflexive, rotund Banach space. It is this space which will be Received by the editors September 29, 1972. AMS (MOS) subject classifications (1970). Primary 41A65; Secondary 46B10.