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

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

Proof

To prove this inequality note it is trivial in the case y = 0. Thus we may assume <y, y> is nonzero. Further, let . Thus we may let

which equals

We now choose

Plugging in for λ we wind up getting

which is true

Taking the square root gives us

Q.E.D.

Notable special cases

• Formulated for Euclidean space Rⁿ, we get

• In the case of square-integrable complex-valued functions, we get

These latter two are generalized by the Hölder inequality.

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

See also

triangle inequality

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