Nonexpansive projections on subsets of Banach spaces 论文

摘要

If C is a convex subset of a Banach space E, a projection is a retraction r of C onto a subset F which for each x e C maps each point of the ray {r(x) + t(x — r(x)): t ^ 0} Π C onto the same point r(x). A retraction r is said to be orthogonal if for each x, x — r(x) is normal to F in a sense related to that of R. C. James. This paper establishes three main results. First, a nonexpansive projection is necessarily an orthogonal retraction; if E is smooth, the converse is also true. Second, if E is smooth then there can exist at most one nonexpan-sive projection of C onto a given subset F. Third, if E is uniformly smooth and there exists a nonexpansive retraction of C onto F, then there exists a nonexpansive projection of C onto F. The proximity mapping is a nonexpansive projection in a Hubert space, but not in a general Banach space. We shall adopt the following conventions throughout this paper: E always denotes a real Banach space, E * its dual space, C a non-empty closed convex subset of E, and F a nonempty closed subset of C. We do not assume that F is convex. A mapping f: C—*C i s s a i d t o b e nonexpansive if \\\\f(x) — f(y) | | ^ \\\\x — y\\ \\ f o r a l l x,yeC. F is said to be a nonexpansive retract of C if there exists a retrac-tion of C onto F which is a nonexpansive mapping. Nonexpansive retracts are of interest because they generalize two results, one linear in reflexive Banach spaces, one nonlinear in Hubert space. First, if E is reflexive and L: E — • E is linear with | |L | | < * 1, then the mean ergodic theorem implies that the means L n = n"