Triangle inequality for norms

Builds on Norm

Statement

Let $V$ be a vector space over $\R$ or $\C$ with a norm $\norm{\cdot}$ . Then, for all $x, y \in V$ , $\norm{x + y} \le \norm{x} + \norm{y} \qquad \text{and} \qquad \bigl| \norm{x} - \norm{y} \bigr| \le \norm{x - y}.$

Remarks

The first inequality is an axiom of a norm; the second, the reverse triangle inequality, follows from it. For the norm induced by an inner product, the first inequality follows from the Cauchy–Schwarz inequality. In $\R^n$ , it says that one side of a triangle is at most the sum of the other two sides.

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Vector space definition
Norm definition
Triangle inequality for norms statement

Tag	`0FJZMBX0ZZ`
URI	`m2://0FJZMBX0ZZ`
Kind	statement, proposition
Writing	W0 stub
MSC	46B20
Status	This copy of the site has no build data. Run `m2 status 0FJZMBX0ZZ`.
Uses	Norm

Triangle inequality for norms

Statement

Remarks

Reading path

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