I was surprised to see a reply to a comment on his answer to a Quora question by a research mathematician claiming that Hilbert spaces were actually due to J. W. Gibbs rather than to D. Hilbert. The Wikipedia entry for Hilbert spaces does not credit Gibbs at all, and the entry for Gibbs does not say enough to lead me to such a conclusion either. It is the case that in the entry for Hilbert spaces, applications to the phenomenon known as Gibbs phenomenon are briefly mentioned, but it’s not clear yet to me that anything in the Wikipedia entry for Gibbs phenomenon suggests that that topic was somehow the motivation for defining Hilbert spaces.

Consequently, in searching online for information about this but historical trivia led me to this site through the question at the following link:

Are there any records that show how Hilbert came to "invent" or "discover" Hilbert spaces?

The answers and comments there don’t seem to suggest either that J. W. Gibbs invented Hilbert spaces, so my question remains.

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    $\begingroup$ Not even Hilbert "invented" Hilbert spaces, Schmidt did $l^2$ and von Neumann the abstract version. Hilbert only defined infinite dot products without the spaces in the context of integral equations, and Riesz later called them "Hilbert" in his honor, see Dieudonne's History of Functional Analysis, VI.1. A reply to a comment to an answer on Quora is not a reliable source. And without a link or quote it is hard to even judge what the commenter might have had in mind. Gibbs introducing dot product of finite-dimensional vectors? $\endgroup$
    – Conifold
    Oct 24, 2023 at 11:17
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    $\begingroup$ @Conifold: Gibbs introducing dot product of finite-dimensional vectors? -- I was completely baffled by the suggestion of Gibbs. In thinking about before 1910, off-hand all I can think of are Fréchet, Volterra, Hadamard. A real stretch might be some other analysts, such as Riesz or Levi or Lebesgue. But Gibbs?? That's not even wrong. However, your observation about dot products does seem like something that might be considered by someone very naive about the subject. $\endgroup$ Oct 24, 2023 at 12:14
  • $\begingroup$ @Conifold, thanks for pointing out the unreliability of Quora as a source as if I didn’t know that. How does it come to be that a research mathematician is so wrong, unless someone somewhere who is credited with reliable authority on the subject is training these people in rumors? Dieudonne would, in my view, be a better source, and it wasn’t one I’d read, so my question got a useful response. $\endgroup$ Nov 8, 2023 at 3:31
  • $\begingroup$ @Conifold, Regarding the role of Hilbert in the founding of the subject that bears his name, it makes sense that the observation that infinite dimensional real and complex vector spaces of functions can be endowed with a bilinear (sesquilinear in the complex case) functional that generalizes the notion of a dot product would be considered pivotal, even if that observation is “restricted” to cases in which the “generalized dot product” is computed via integration (convergent weighted infinite series in the case of sequence, or “signal”, spaces). $\endgroup$ Nov 8, 2023 at 3:43


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