Cauchy–Schwarz inequality

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. 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. The inequality for sums was published by Augustin-Louis Cauchy (1821), while the corresponding inequality for integrals was first proved byViktor Bunyakovsky (1859). The modern proof of the integral inequality was given by Hermann Amandus Schwarz (1888). The Cauchy–Schwarz inequality states that for all vectors u {displaystyle u} and v {displaystyle v} of an inner product space it is true that where ⟨ ⋅ , ⋅ ⟩ {displaystyle langle cdot ,cdot angle } 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 Moreover, the two sides are equal if and only if u {displaystyle mathbf {u} } and v {displaystyle mathbf {v} } are linearly dependent (meaning they are parallel: one of the vector's magnitudes is zero, or one is a scalar multiple of the other). If u 1 , … , u n ∈ C {displaystyle u_{1},ldots ,u_{n}in mathbb {C} } and v 1 , … , v n ∈ C {displaystyle v_{1},ldots ,v_{n}in mathbb {C} } , and the inner product is the standard complex inner product, then the inequality may be restated more explicitly as follows (where the bar notation is used for complex conjugation):