Although special cases of the Cauchy–Schwarz inequality, also known simply as Cauchy's inequality, Schwarz's inequality or Bunyakovsky's inequality, had long been used in mathematical proofs, the first formulation accompanied by a proof—and thus treating the inequality as a result of interest in its own right—was due to Cauchy.
Cauchy and the numerical inequality
-------------------------------
Augustin-Louis Cauchy was the first to tackle the subject, in Note 2 of his Analyse géométrique (Geometric Analysis), based on his course at the École polytechnique and published in 1821. The note is entitled Sur les formules qui résultent de l’emploi du signe > ou <, et sur les moyennes entre plusieurs quantités (On formulas resulting from the use of the signs > and <, and on means of several quantities).
After stating numerous obvious or classical inequalities, the fifteenth theorem asserts:
Let a, a′, a″… be any n quantities. If these quantities are not all equal, the numerical value of the sum a + a′ + a″+… will be less than the product so that:
