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Parallelogram Law

Let $\vert\cdot\vert$ denote the Norm of a quantity. Then the quantities $x$ and $y$ satisfy the parallelogram law if

\begin{displaymath}
\Vert x+y\Vert^2+\Vert x-y\Vert^2 = 2\Vert x\Vert^2+2\Vert y\Vert^2.
\end{displaymath}

If the Norm is defined as $\vert f\vert = \sqrt{\left\langle{f\vert f}\right\rangle{}}$ (the so-called L2-Norm), then the law will always hold.

See also L2-Norm, Norm




© 1996-9 Eric W. Weisstein
1999-05-26