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?
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$\begingroup$ 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. $\endgroup$– Jack D'AurizioCommented Apr 20, 2017 at 16:11
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$\begingroup$ Note that the Cauchy-Schwarz inequality is very important in the study of mathematical statistics. $\endgroup$– Michael R. ChernickCommented Apr 20, 2017 at 17:02
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1 Answer
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An inner product was not defined until ~1905 by Hilbert. Orthogonality of the trigonometric functions in the Fourier expansion was experimentally discovered in the late 1700's, and Fourier built on this work with more general orthogonal function expansions. Cauchy only came up with his inequality for complex Euclidean space in the 1820's (I think that's right.) And no connection was made between orthogonality of functions with respect to an integral and the Euclidean dot product until the second half of the 1800's. Part of the reason for this disconnect may have been that there was no natural path from the sum to the integral because the Riemann integral was not defined until several decades later.
A note by Bunyakowsky appeared in a journal in 1859 that the discrete case could be generalized to integrals, but it was ignored because no applications were given. It wasn't until the next decade that Schwarz published a paper on minimal surfaces where the Schwarz inequality for the integral was rediscovered and used to measure something like a distance in order to get at a solution for PDEs. But, even then, it was not noticed that there was a general concept of a "norm" or "distance function." Cantor began developing set axioms for the foundation of Mathematics about that time, and by the turn of the 20th century, people started thinking of abstract "spaces" where a point could be an object such as a function. And that's what led to Hilbert's definition of a general inner product space, and to a great deal of modern Mathematics. The current version of the Cauchy-Schwarz inequality was proved in this context.
So, the Schwarz inequality for integrals came a few decades before the definition of an inner product. And Cauchy's inequality for the discrete case came about 80 years earlier. I believe it was Hilbert who tagged the general inner product inequality as Cauchy-Schwarz because of the work of these two Mathematicians.
answered Apr 20, 2017 at 16:41
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1$\begingroup$ That 1859 date is (also?) the date of Bunyakowsky's publication on this, I think. $\endgroup$ Commented Apr 20, 2017 at 16:46
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$\begingroup$ @paulgarrett : Yes, that was the date, and I would have recorded the name if I could have remembered it. Thanks. I'll put that in now. $\endgroup$ Commented Apr 20, 2017 at 16:53
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1$\begingroup$ This is very interesting historical information. $\endgroup$ Commented Apr 20, 2017 at 17:02
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1$\begingroup$ I wonder how they were thinking to integrals without any Riemann sum. As a well-defined linear operator on polynomials and power series ? $\endgroup$– reunsCommented Apr 20, 2017 at 19:23
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1$\begingroup$ Fantastic answer. Where did you learn all this? $\endgroup$ Commented Apr 21, 2017 at 0:06