The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality for integrals was published by and . Schwarz gave the modern proof of the integral version. Statement of the inequality 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. Every inner product gives rise to a norm, called the or , where the norm of a vector is denoted and defined by: so that this norm and the inner product are related by the defining condition where is always a non-negative real number (even if the inner product is complex-valued). By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form: Moreover, the two sides are equal if and only if and are linearly dependent. Special cases Sedrakyan's lemma - Positive real numbers Sedrakyan's inequality, also called Bergström's inequality, Engel's form, the T2 lemma, or Titu's lemma, states that for real numbers and positive real numbers : It is a direct consequence of the Cauchy–Schwarz inequality, obtained by using the dot product on upon substituting and . This form is especially helpful when the inequality involves fractions where the numerator is a perfect square. R2 - The plane The real vector space denotes the 2-dimensional plane. It is also the 2-dimensional Euclidean space where the inner product is the dot product. If and then the Cauchy–Schwarz inequality becomes: where is the angle between and . The form above is perhaps the easiest in which to understand the inequality, since the square of the cosine can be at most 1, which occurs when the vectors are in the same or opposite directions. It can also be restated in terms of the vector coordinates , , , and as where equality holds if and only if the vector is in the same or opposite direction as the vector , or if one of them is the zero vector. Rn - n-dimensional Euclidean space In Euclidean space with the standard inner product, which is the dot product, the Cauchy–Schwarz inequality becomes: The Cauchy–Schwarz inequality can be proved using only