The Cauchy-Bunyakovsky inequality and its mathematical interpretations

Bunyakovsky’s integral inequality (1859) is one of the familiar tools of modern Analysis. We try and understand what Bunyakovsky did, why he did it, why others did not follow the same path, and how his inequality was interpreted. Our results are the following. Close reading of his paper shows that it was for him an outgrowth of his interest in means, already apparent in Cauchy’s work (1821), but in the context of Probability and Statistics. Other theories such as the method of least squares, the theory of generalized Fourier expansions, and the notion of orthogonal projection, that now belong to the same circle of ideas, led to closely related results, but not to Bunyakovsky’s inequality (Bessel, 1828; Liouville, 1836; Grassmann, 1862). By relating the result to quadratic forms, Schwarz (1885) opened the way to a geometric interpretation of the inequality that became important in the theory of integral equations. At about the same time, the Rogers-Hölder inequality suggested generalizations of Cauchy’s and Bunyakovsky’s results in an entirely different direction. Later extensions and reinterpretations show that no single result, even now, subsumes all known generalizations.