Bulletin of the
Korean Mathematical Society BKMS

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