Rongzhou at 21:32, 30 September 2026

2026-09-30T21:32:13Z

← Older revision	Revision as of 21:32, 30 September 2026
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		== Euclidean geometry ==

		In Euclidean geometry, the triangle inequality states that the sum of the lengths of two sides of a triangle is always larger than the third side.

		For instance, in $\triangle ABC$

		[[File:Triangle.png\|250px\|center]]

		$$
		a+b>c \\
		a+c>b \\
		b+c>a
		$$

		== Generalisation to vector spaces ==

		Let $V$ be an [[inner product space]] over $\mathbb{F}$.

		$$\forall u,v\in V,\; \left\\| u+v \right\\| \le \left\\| u \right\\| + \left\\| v \right\\|.$$

		'''Proof''':

		$$
		\begin{aligned}
		\left\\| u+v \right\\|^2
		&=\langle u+v,u+v\rangle
		=\langle u,u\rangle+\langle v,v\rangle+\langle u,v\rangle+\langle v,u\rangle\\
		&=\langle u,u\rangle+\langle v,v\rangle+\langle u,v\rangle+\overline{\langle u,v\rangle}
		=\left\\| u \right\\|^2+\left\\| v \right\\|^2+2\operatorname{Re}\langle u,v\rangle,\\
		\text{Note }\operatorname{Re}\langle u,v\rangle
		&\le\|\langle u,v\rangle\|
		\stackrel{\text{Cauchy-Schwarz}}{\le}\left\\| u \right\\|\left\\| v \right\\|,\\
		\left\\| u \right\\|^2+\left\\| v \right\\|^2+2\operatorname{Re}\langle u,v\rangle
		&\le\left\\| u \right\\|^2+\left\\| v \right\\|^2+2\left\\| u \right\\|\left\\| v \right\\|
		=(\left\\| u \right\\|+\left\\| v \right\\|)^2,\\
		\Rightarrow \left\\| u+v \right\\|^2 \leq (\left\\| u \right\\|+\left\\| v \right\\|)^2,\\
		\because \left\\| . \right\\| \geq 0 \\
		\therefore \left\\| u+v \right\\|&\le\left\\| u \right\\|+\left\\| v \right\\|
		\end{aligned}
		$$

Rongzhou: Created the wiki page for triangle inequality

2026-09-30T21:25:04Z

Created the wiki page for triangle inequality

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