Linear and multilinear algebra; matrix theory Inner product spaces Theorem

Cauchy–Schwarz inequality

Builds on Inner product space Norm induced by an inner product

Statement

Let $V$ be an inner product space over $\R$ or $\C$ , with the induced norm $\norm{\cdot}$ . Then, for all $x, y \in V$ , $\abs{\langle x, y \rangle} \le \norm{x} \norm{y}.$ Equality holds if and only if $x$ and $y$ are linearly dependent.

Remarks

For the dot product on $\R^n$ , it says $\left(\sum_i x_i y_i\right)^2 \le \left(\sum_i x_i^2\right)\left(\sum_i y_i^2\right)$ . In a real inner product space, it allows the definition of the angle $\theta \in [0, \pi]$ with $\cos\theta = \langle x, y \rangle / (\norm{x} \norm{y})$ for nonzero $x$ and $y$ . One proof uses that $\norm{x - ty}^2 \ge 0$ for all real $t$ , so a quadratic has a nonpositive discriminant. It implies the triangle inequality for the induced norm.

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Vector space definition
Inner product definition
Norm definition
Inner product space definition
Norm induced by an inner product definition
Cauchy–Schwarz inequality statement

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Tag	`0RFWW56Y22`
URI	`m2://0RFWW56Y22`
Kind	statement, theorem
Writing	W1 stated
MSC	15A63, 46C05
Status	This copy of the site has no build data. Run `m2 status 0RFWW56Y22`.
Uses	Inner product space, Norm induced by an inner product
Same as	Wikidata Q190546 Wikipedia Cauchy–Schwarz inequality

Cauchy–Schwarz inequality

Statement

Remarks

Reading path

Where this leads

Further reading

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