nLab Cauchy–Schwarz inequality

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Contents

Context

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

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Basic facts

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Functional analysis

Contents

Statement

Consider an inner product space in the sense of a vector space 𝒱\mathscr{V} equipped with a Hermitian form ,\langle -,- \rangle which is positive semi-definite:

v𝒱v,v0. v \in \mathscr{V} \;\;\; \vdash \;\;\; \langle v , v \rangle \;\geq 0\; \,.

(we do not need to assume positive definiteness, cf. MO:a/2548691, Ćurgus)

then for all pairs of vectors v,wVv,w \,\in\, V the following Cauchy-Schwarz inequality holds:

|u,v| 2u,uv,v. \left\vert\langle u,v\rangle\right\vert^2 \;\leq\; \langle u,u \rangle \cdot \langle v,v \rangle \,.

In terms of the norm ,\Vert-\Vert \,\coloneqq\, \sqrt{\langle -,-\rangle} and the absolute value this means equivalently:

|u,v|uv. {\big\vert \langle u,v\rangle \big\vert} \;\leq\; \left\Vert u \right\Vert \cdot \left\Vert v \right\Vert \,.

References

Original proofs are due to Cauchy in 1821, Bouniakowsky in 1859, Hermann Schwarz in 1888.

Review:

Last revised on January 14, 2024 at 09:12:13. See the history of this page for a list of all contributions to it.