nLab Cauchy–Schwarz inequality

Statement

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

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

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

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

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

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

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

Related concepts

• GNS-construction

References

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

Review:

• Branko Ćurgus, Cauchy-Bunyakovsky-Schwarz inequality

• Wikipedia, Cauchy-Schwarz inequality