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In mathematics, the Cauchy-Schwarz inequality, also known as the Schwarz inequality, or the Cauchy-Bunyakovski-Schwarz inequality, 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.

1 Proof

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

and it follows that

multiplying out, the result follows.

2 Notable special cases

Formulated for Euclidean space Rⁿ, we get

In the case of square-integrableIn mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. The set of all measurable functions that are square-integ complex-valued functionIn mathematics, a function is a relation such that each element of a set (the domain is associated with a unique element of another (possibly the same) set (the codomain not to be confused with the range . The concept of a function is fundamental to virtus, we get

These latter two are generalized by the Hölder inequality.

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

3 See also

triangle inequalityIn mathematics, the triangle inequality is a statement which states roughly that the distance from A to B to C is never shorter than going directly from A to C. The triangle inequality is a theorem in spaces such as the real numbers, Euclidean space, Lp s inner product space for a proof of the Cauchy-Schwarz inequality.

Inequalities

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