Cauchy–Schwarz inequality

In mathematics, the Cauchy-Schwarz inequality, also known as the Schwarz inequality, or the Cauchy-Bunyakovski-Schwarz inequality, named after Augustin Louis Cauchy, Hermann Amandus Schwarz and Viktor Yakovlevich Bunyakovsky, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied to infinite series and integration of products, and in probability theory, applied to variances and covariances. The inequality states that if x and y are elements of real or complex inner product spaces then

$|\langle x,y\rangle |^{2}\leq \langle x,x\rangle \cdot \langle y,y\rangle .$

The two sides are equal if and only if x and y are linearly dependent.

An important consequence of the Cauchy-Schwarz inequality is that the inner product is a continuous function.

Another form of the Cauchy-Schwarz inequality is given using the notation of norm, as explained under norms on inner product spaces, as

$|\langle x,y\rangle |\leq \|x\|\cdot \|y\|.\,$

Proof

As the inequality is trivially true in the case y = 0, we may assume <y, y> is nonzero. Let $\lambda \in \mathbb {C}$. Then,

$0\leq \left\|x-\lambda y\right\|^{2}=\langle x-\lambda y,x-\lambda y\rangle$

$=\langle x,x\rangle -\lambda \langle x,y\rangle -{\overline {\lambda }}\langle y,x\rangle +|\lambda |^{2}\langle y,y\rangle .$

Choosing

$\lambda =\langle y,x\rangle \cdot \langle y,y\rangle ^{-1}$

we obtain

$0\leq \langle x,x\rangle -|\langle x,y\rangle |^{2}\cdot \langle y,y\rangle ^{-1}$

which is true if and only if

$|\langle x,y\rangle |^{2}\leq \langle x,x\rangle \cdot \langle y,y\rangle$

or equivalently:

${\big |}\langle x,y\rangle {\big |}\leq \left\|x\right\|\left\|y\right\|.$

Q.E.D.

Notable special cases

• In the case of the Euclidean space Rⁿ, we get

$\left(\sum _{i=1}^{n}x_{i}y_{i}\right)^{2}\leq \left(\sum _{i=1}^{n}x_{i}^{2}\right)\left(\sum _{i=1}^{n}y_{i}^{2}\right).$

• For the inner product space of square-integrable complex-valued functions, one has

$\left|\int f^{(}x)g(x)\,dx\right|^{2}\leq \int \left|f(x)\right|^{2}\,dx\cdot \int \left|g(x)\right|^{2}\,dx.$

A generalization of these two inequalities is the Hölder inequality.

• A notable strengthening of the basic inequality occurs in dimension n = 3, where a stronger equality holds:

$\langle x,x\rangle \cdot \langle y,y\rangle =|\langle x,y\rangle |^{2}+|x\times y|^{2}.$

See also

• triangle inequality

• inner product space for a proof of the Cauchy-Schwarz inequality.