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 ∈ C maps each point of the ray {r(x) + t(x−r(x)) : t ≧ 0}∩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
nonexpansive 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 Hilbert space, but not in a general Banach
space.
