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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 30th 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded a statement. <a href="https://ncatlab.org/nlab/revision/diff/Cauchy%E2%80%93Schwarz+inequality/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cauchy%E2%80%93Schwarz+inequality/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Cauchy%E2%80%93Schwarz+inequality">current</a>

   Added a statement.

   diff, v2, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 30th 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: Given a real or complex [[inner product space]], we have $$|\langle u,v\rangle|\le \|u\|\cdot\|v\|.$$ ## Names 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. ## References * Wikipedia, _[Cauchy-Schwarz inequality](https://en.wikipedia.org/wiki/Cauchy-Schwarz_inequality)_ [[!redirects Cauchy-Schwarz inequality]] [[!redirects Cauchy inequality]] <a href="https://ncatlab.org/nlab/revision/diff/Cauchy%E2%80%93Schwarz+inequality/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cauchy%E2%80%93Schwarz+inequality/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Cauchy%E2%80%93Schwarz+inequality">current</a>

   Added:

   Given a real or complex inner product space, we have

   $$|\langle u,v\rangle|\le \|u\|\cdot\|v\|.$$

   Names

   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.

   References

   • Wikipedia, Cauchy-Schwarz inequality

   !redirects Cauchy-Schwarz inequality !redirects Cauchy inequality

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded cross-link with *[[Augustin Cauchy]]* <a href="https://ncatlab.org/nlab/revision/diff/Cauchy%E2%80%93Schwarz+inequality/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cauchy%E2%80%93Schwarz+inequality/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Cauchy%E2%80%93Schwarz+inequality">current</a>

   added cross-link with Augustin Cauchy

   diff, v3, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 14th 2024
   • (edited Jan 14th 2024)
   • PermaLink
   Author: Urs
   Format: MarkdownItexTouched 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](https://math.stackexchange.com/a/2548691/58526) 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](https://ncatlab.org/nlab/show/Cauchy–Schwarz+inequality#Ćurgus)). <a href="https://ncatlab.org/nlab/revision/diff/Cauchy%E2%80%93Schwarz+inequality/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cauchy%E2%80%93Schwarz+inequality/7">v7</a>, <a href="https://ncatlab.org/nlab/show/Cauchy%E2%80%93Schwarz+inequality">current</a>

   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).

   diff, v7, current