Vol. 47, No. 2, 1973

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Nonexpansive projections on subsets of Banach spaces

Ronald Elroy Bruck, Jr.

Vol. 47 (1973), No. 2, 341–355

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(xr(x)) : t 0}∩C onto the same point r(x). A retraction r is said to be orthogonal if for each x, xr(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.

Mathematical Subject Classification 2000
Primary: 47H99
Secondary: 46B99
Received: 14 June 1972
Published: 1 August 1973
Ronald Elroy Bruck, Jr.