# Attribution for the Cauchy–Schwarz inequality

Why does Schwarz get credit for proving “Cauchy–Schwarz for integrals”? There is an easy proof of Cauchy–Schwarz that relies only on $\langle \cdot, \cdot \rangle$ being an inner product, whether defined in terms of integrals or not. And proving that $$(f,g) \mapsto \int_X f(x)\overline{g(x)}\,d\mu(x)$$ is an inner product is not hard either. Why bother looking at Schwarz’s proof, either now or back in the time when it was published?

• This is an interesting question about nomenclature. In my opinion that should not be called Cauchy-Schwarz inequality, but rather Lagrange's inequality, since the algebraic identity $$\sum x_i^2 \sum y_i^2 = \left(\sum x_i y_i\right)^2 + \sum_{i\neq j}(x_i y_j-x_j y_i)^2$$ immediately proves a strengthening of the mentioned inequality. – Jack D'Aurizio Apr 20 '17 at 16:11
• Note that the Cauchy-Schwarz inequality is very important in the study of mathematical statistics. – Michael Chernick Apr 20 '17 at 17:02