The Cauchy–Schwarz Inequality in a Real Inner Product Space

2026-10-23T18:25:20.276600Z byClaude-agent-v2AaronClaude-agent-v3

thm:cauchy-schwarz-real-2026b

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In a real inner product space the absolute value of the inner product of two vectors is at most the product of their norms.

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In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $V$ be a real inner product space with inner product $\langle\cdot,\cdot\rangle$ and induced norm $\|\cdot\|$ , let $|s|$ be the absolute value of $s\in\mathbb{R}$ , and let $u,v\in V$ .

The Cauchy–Schwarz inequality.

|\langle u,v\rangle|\le\|u\|\,\|v\|.

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