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 A metric characterization of Hilbert spaces Bull. Korean Math. Soc. 1996 Vol. 33, No. 1, 35-38 Jin Sik Mok Sun Moon University Abstract : Suppose that {\bf X} is a real or complex Banach space with norm $| \cdot |$. Then {\bf X} is not a Hilbert space if and only if there are four points $x$, $x^\prime$, $y$, and $y^\prime$ in {\bf X} such that $|x|= |x^\prime|$, $|y| = |y^\prime|$, $|x - y| < |x^\prime - y^\prime|$, and $|x+y| < |x^\prime + y^\prime|$. Keywords : Parallelogram identity, Hilbert space MSC numbers : 46C15 Downloads: Full-text PDF