Strictly convex spaces via semi-inner-product space orthogonality

Let (X, 11 * 11) be a normed space, and let [* * be any semi-inner-product on it. We show that (X, 11 11) iS strictly convex if and only if IIy+zII > IIYII whenever [z, y] =0 and z$0, and if and only if [Ax, x].0 whenever llI+All 1 and Ax$0. The condition that [z, y] =0 can be replaced by a stronger or weaker condition. A complex or real normed linear space (X, II fI) is strictly convex if each point of the unit sphere is an extreme point of the unit ball. Every normed space has at least one semi-inner-product [4, Theorem 2, p. 31], i.e., a map [., * ] on XXX to C (resp., to R) such that (i) [Xx +y, z ] =x [x, z] + [y, z], (ii) [x,x]= IxiI2, (iii) I [x, y] ? 1xjjI1IyII, for all x, y, z in X, X in C (resp., X in R). For a given semi-inner-product [, * ] on the space, one can say that y is orthogonal to z if [z, y ] = 0; the condition that y is orthogonal to z then depends on the choice of semi-inner-product. Nonetheless, we show that if [-, *] is any semi-inner-product on (X, II -II), the space is strictly convex if and only if IIy+zII>IIyII whenever y is orthogonal to z#O, and if and only if x is never orthogonal to Axf O for operators A such that II I+AII < 1. This result still holds if the original orthogonality is replaced by a stronger or weaker form, both of which depend only on the normed space, and the latter of which is equivalent to that of James [3, p. 265]. The last paragraph contains an application of condition (iv). Related results have been obtained by Berkson [1, Theorem 5.1, p. 381, and Lemma 5.3, p. 382], James [3, Theorem 4.3, p. 275, and Theorem 5.2, p. 279], and Palmer [5, p. 4]. Presented to the Society, October 25, 1969 under the title Some characterizations of strictly convex Banach spaces; received by the editors December 23, 1969. AMS 1969 subject classifications. Primary 4610.