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- W2069819507 abstract "A normed space E is an inner product space if and only if for every 2-dimensional subspace V and every segment I c V, the corresponding metric projections satisfy the commutative property PIPV = PVPI. For a subset A of a normed linear space E, we denote by PA the metric projection on A, i.e. the set-valued mapping which corresponds to each x E E the (possibly empty) set of its best approximations in A: PA x = { y E A; lIx y = d(x, A)). A is called proximinal if PA X is nonempty for every x E E. If E is a Hilbert space and A is a closed subspace of E, then PA is just the (single-valued) orthogonal projection onto A. There are several known characterizations of inner product spaces which can be stated in terms of the metric projections. See e.g. [3], [5], [8], and [9]. We shall consider three other such conditions below. For all of these three characterizing conditions, the necessity part is immediate. The weakest condition (hence strongest characterization) is due to Lorch [7] (cf. also Day [1, p. 152]): (L) If IlxII = IIyII = 1 and A = {/8x + y//3,; /i 3 0 real), then x + y E PAO' The next condition, due to Gurari and Sozonov [4], is: (GS) If llxll = Ilyll = 1 and A is the segment [x, y] = {ax + (I a)y; 0 3) is due to Joichi [6]: (J) If V is a 2-dimensional subspace of E, u E E with 0 E Pvu and A = SO is the unit circle in V: So = {v E V: lvII = 1), then PAu = A. We first give an easy proof that (J) X (GS) X (L), and hence this yields an alternate approach to the more involved sufficiency proofs as given in [4] and [6]. (J) =X> (GS): We may assume dim E = 3. If (GS) fails, ,then there exists x, y in E with lxii = iIyH = 1 and 0 < y < such that the element z = yx + (1 y)y satisfies lizil = min0<x<l1121X + (1 2)yi < II2(x +Y)II Extend the segment [x, y] to a supporting hyperplane V + z to the ball {w E E: Received by the editors July 25, 1977. AMS (MOS) subject classifications (1970). Primary 46B99; Secondary 41A65." @default.
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- W2069819507 date "1978-01-01" @default.
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- W2069819507 title "Another approximation theoretic characterization of inner product spaces" @default.
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- W2069819507 doi "https://doi.org/10.1090/s0002-9939-1978-0495846-6" @default.
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