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arrow_back Does the Pythagorean theorem generalize to arbitrary inner product spaces?

by SIlver Status
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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 SIlver Status
(18,455 points)

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