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I think it's fuzzy as to whether or not this question is appropriate to ask on this site.

The reason I ask it that the characteristics of Hilbert spaces are very much used in expressing quantum mechanics.

I am wondering if Hilbert had this in mind all along or was it just a fortunate accident for the early pioneers of quantum mechanics that he developed this when he did? Thank you

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    $\begingroup$ He did not have QM in mind (this was 20 years before it), but it was not an accident either. And he did not give the modern abstract definition (von Neumann did that, and he did have QM in mind) but rather examples of inner products for functions and sequences. Wikipedia has a detailed history section. $\endgroup$
    – Conifold
    Commented Dec 3, 2017 at 21:34

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From Steen’s paper mentioned in a very similar MO question just yesterday:

Hilbert himself was astonished that the spectra of his quadratic forms should come to be interpreted as atomic spectra. “I developed my theory of infinitely many variables from purely mathematical interests, and even called it ‘spectral analysis’ without any presentiment that it would later find an application to the actual spectrum of physics.”

”Records of how he came to invent” his space (originally just $l^2$) are his papers, quoted by Steen.

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  • $\begingroup$ Interesting answer ...wow......this is hard to get my head around. Physics is filled with these "accidents" from math and visa versa. $\endgroup$
    – Sedumjoy
    Commented Dec 4, 2017 at 0:52
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The notion of Hilbert space comes from Hilbert's theory of integral equations. Of course, it was partially motivated by physics, by the theory of oscillations in classical mechanics, but this theory was developed much earlier than quantum mechanics: the main work Grundzuge einer allgemeinen Theorie der linearen Intergalgleichungen (Foundations of the general theory of linear integral equations), in 6 parts was published in 1904-1910. Hilbert's own exposition (besides original papers) is in Courant Hilbert, Methods of mathematical physics. The very title shows that he definitely was motivated by physics, while quantum mechanics was not discovered yet.

There are many anecdotes showing that Hilbert was surprised by applications to quantum mechanics and by the very term "Hilbert space". Once in a conference, he asked the speaker: "What is a Hilbert space?"

W. Feller was fond of repeating: "I could stare all my life at a symmetric matrix, I will never get Hilbert space out of it!" (Cited from G-C Rota, Indiscrete thoughts p. 223).

There are many other examples when mathematicians develop new tools just in time, shortly before they find new applications in physics (Riemannian geometry and tensor calculus, Hamiltonian formalism (wave-particle duality) etc.) On the other hand, this mathematical research is motivated by already existing physics, at least partially.

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    $\begingroup$ your exposition is outstanding...I already accepted the above answer but voted up on yours.....Very good research for me to ponder on. Thank you $\endgroup$
    – Sedumjoy
    Commented Dec 4, 2017 at 15:49

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