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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 30th 2022

    Added a statement.

    diff, v2, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 30th 2022

    Added:

    Given a real or complex inner product space, we have

    |u,v|uv.|\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

    !redirects Cauchy-Schwarz inequality !redirects Cauchy inequality

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2022

    added cross-link with Augustin Cauchy

    diff, v3, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 14th 2024
    • (edited Jan 14th 2024)

    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