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Does the Pythagorean theorem generalize to arbitrary inner product spaces?
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The Pythagorean theorem generalizes naturally to arbitrary inner product spaces.
Theorem $1 .$ If $\mathrm{x} \perp \mathrm{y}$, then
$$\|\mathrm{x}+\mathbf{y}\|^{2}=\|\mathrm{x}\|^{2}+\|\mathbf{y}\|^{2}$$
Proof. Suppose $\mathbf{x} \perp \mathbf{y}$, i.e. $\langle\mathbf{x}, \mathbf{y}\rangle=0$. Then
$$\|\mathrm{x}+\mathbf{y}\|^{2}=\langle\mathrm{x}+\mathbf{y}, \mathrm{x}+\mathbf{y}\rangle=\langle\mathrm{x}, \mathrm{x}\rangle+\langle\mathbf{y}, \mathrm{x}\rangle+\langle\mathrm{x}, \mathbf{y}\rangle+\langle\mathbf{y}, \mathbf{y}\rangle=\|\mathrm{x}\|^{2}+\|\mathbf{y}\|^{2}$$
as claimed.
by Diamond (58,473 points)

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