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I. Gurarii and N. I. Gurarii, Bases in uniformly convex and uniformly flattened Banaeh spaces, Math. USSR Izvestija, 5 (1971), 220-225. J. R. Holub, On the metric geometry of ideals of operators on Hilbert space, Math. Ann. 201 (1973), 157-163. R. C. James, Orthogonality in normed linear spaces, Duke Math. , 12 (1945), 291-302. R. C. James, Bases and reflexivity of Banach spaces, Ann. , 52 (1950), 518-527. S. Kerlin, Bases in Banach spaces, Duke Math. , 15 (1948), 971-985. W. A. Kirk, A fixed point theorem for mappings which do not increase distance, Amer.

Let X be smooth and let x -" fx be ~ s u p p o r t map of X I[0} to X* I{0}. ~ , z 6 X. to (ii)) Let 0 # x E X. By right uniqueness Suppose (we've already x 9 y and x ~ z where proved (i) is equivalent the unique ~ such that x z (~x + y) must be 0 and likewise the unique ~ such that x • (Sx + z). that fx(y) = 0 = fx(Z) (see Observation fx(y + z) = fx(y ) + f(z) = O. (iii) implies (ii). In each case (i)). By Theorem though this yields Thus 3 we get x z y + z. Suppose orthogonality in X is right additive, and let 0 # x 6 X be such that x - (~x + y) and x - (Sx + y).

M. S. Brodskii and D. P. Milman [6]. Browder [8] and M. Edelstein [24]. E. Theorem 3 is basically due to A 49 W. A. Kirk [35], while Corollary i had been discovered earlier by F. E. Browder [8]. Theorem 5 is due to F. E. Browder [8]. A class of strictly convex spaces which have the property that every closed hounded convex subset has normal structure was introduced by A. L. Garkavi [27] for the purpose of characterizing those spaces for which v every bounded set has at most one Cebysev center. This class goes by the name of spaces uniformly convex in every direction; a normed linear space X is uniformly convex in every direction (UCED) whenever given sequences (Xn), (yn) c_ S(X) and given z E X ~ [ 0 } for which Xn " Yn = ~n z and .