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In mathematics, the Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics.[1]

The inequality for sums was published by Augustin-Louis Cauchy (1821), while the corresponding inequality for integrals was first proved by Viktor Bunyakovsky (1859). The modern proof of the integral inequality was given by Hermann Amandus Schwarz (1888).[1]

Statement of the inequality[edit]

The Cauchy–Schwarz inequality states that for all vectors and of an inner product space it is true that

where is the inner product. Examples of inner products include the real and complex dot product, see the examples in inner product. Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as[2][3]

Moreover, the two sides are equal if and only if and are linearly dependent (meaning they are parallel, one of the vector's magnitudes is zero, or one is a scalar multiple of the other).[4]:14

If and have an imaginary component, the inner product is the standard complex inner product, where the bar notation is used for complex conjugation, and then the inequality may be restated more explicitly as

or

Proofs[edit]

First proof[edit]

Let and be arbitrary vectors in a vector space over with an inner product, where is the field of real or complex numbers. We prove the inequality

and that equality holds if and only if either or is a multiple of the other (which includes the special case that either is the zero vector).

If , it is clear that we have equality, and in this case and are also linearly dependent, regardless of , so the theorem is true. Similarly if . We henceforth assume that is nonzero.

Let

Then, by linearity of the inner product in its first argument, one has

Therefore, is a vector orthogonal to the vector (Indeed, is the projection of onto the plane orthogonal to .) We can thus apply the Pythagorean theorem to

which gives

and, after multiplication by and taking square root, we get the Cauchy–Schwarz inequality. Moreover, if the relation in the above expression is actually an equality, then and hence ; the definition of then establishes a relation of linear dependence between and . On the other hand, if and are linearly dependent, then there exists such that (since ). Then

This establishes the theorem.

Second proof[edit]

Let and be arbitrary vectors in a vector space with an inner product, where is the field of real or complex numbers.

In the special case the theorem is trivially true. Now assume that . Let be given by , then

Therefore, , or .

If the inequality holds as an equality, then , and so , thus and are linearly dependent. On the other hand, if and are linearly dependent, then , as shown in the first proof.

More proofs[edit]

There are indeed many different proofs[5] of the Cauchy–Schwarz inequality other than the above two examples.[1][3] When consulting other sources, there are often two sources of confusion. First, some authors define ⟨⋅,⋅⟩ to be linear in the second argument rather than the first. Second, some proofs are only valid when the field is and not .[6]

Special cases[

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