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Added:
Given a real or complex inner product space, we have
$|\langle u,v\rangle|\le \|u\|\cdot\|v\|.$Known as “Cauchy inequality”, “Cauchy–Schwarz inequality”, “Cauchy–Bouniakowsky–Schwarz” inequality.
Proofs were published by Cauchy in 1821, Bouniakowsky in 1859, Hermann Schwarz in 1888.
!redirects Cauchy-Schwarz inequality !redirects Cauchy inequality
added cross-link with Augustin Cauchy
Touched the entry to make it clear that the inner products to which the Schwarz inequality applies do not need to be positive definite, it suffices for them to be positive semi-definite.
For the moment I am referencing MO:a/2548691 for this point. All published references that I checked so far assume positive-definiteness. (The statement, but not the proof, is also in the course notes Ćurgus).
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