The Pythagorean Theorem in N-Space

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). The Pythagorean Theorem in N-Space

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The Pythagorean Theorem in N-Space

In 2D, the Pythagorean Theorem says that when $x$ and $y$ are orthogonal, as in Fig. 6.8, (i.e., when the triangle formed by $x$, $y$, and $x+y$, with $y$ translated to the tip of $x$, is a right triangle), then we have

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This relationship generalizes to $N$ dimensions, as we can easily show:
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If $x\, then $\ and the Pythagorean Theorem $\ holds in $N$ dimensions. If, on the other hand, we assume the Pythagorean Theorem holds, then since all norms are positive unless $x$ or $y$ is zero, we must have $\. Finally, if $x$ or $y$ is zero, the result holds trivially.

Note that we also have an alternate version of the Pythagorean theorem:

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