Let $X$ be a normed linear space over $\mathbb C$ such that $||x-y|| \ge \dfrac 12 (||x||+||y||)\bigg|\bigg| \dfrac x{||x||}- \dfrac y {||y||} \bigg|\bigg| , \forall 0\ne x, y \in X$ , then is it true that the norm on $X$ comes from an inner-product ? ( I can show that for a complex inner-product space , the inequality is true ) If not true in general , what if we moreover assume $X$ is Banach or finite dimensional ?
2026-04-29 07:34:57.1777448097
A characterization of inner product spaces ?
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Initially, this was proved in
W.A. Kirk and M.F. Smiley, Mathematical Notes: Another characterization of inner product spaces, Amer. Math. Monthly 71 (1964), no. 8, 890–891.
but I don't have access to this paper. If you are interested in the complete history on this question see
F. Dadipour, A. Maric, M. S. Moslehian, and R. Rajic, A glimpse at the Dunkl-Williams inequality Banach J. Math. Anal. Volume 5, Number 2 (2011), 138-151.