# What did Hilbert do on Hilbert spaces to deserve his name?

This question is just curiosity. When I had my first course in Functional Analysis, most of basic theorems about Banach spaces were presented to me as attributed to Banach (Hahn-Banach, Banach-Steinhaus, Banach-Schauder, Banach fixed point ...). And this was enough, for me, to be convinced that the name Banach space is fair. On the other hand, I was surprised that apparently no general result about Hilbert spaces is attributed to Hilbert. Even more, as far as I know, the two most basic theorems, the projection theorem and the representation theorem, are both due to Riesz. Of course, Hilbert studied some particular cases with Schmidt (Hilbert-Schmidt operators/norm), but it seems to me that it is not really fair to call these spaces, Hilbert spaces (maybe Hilbert-Schmidt spaces?). I am probably missing something and then I would like to know the answer to the following

Question: What did Hilbert do about Hilbert spaces to deserve his name?

According to wikipedia's article http://en.wikipedia.org/wiki/Hilbert_space, the name Hilbert space was coined by von Neumann; but it is not clear why and in what year.

Thank you in advance for the historical clarification,

Valerio

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Some information on this at Wikipedia en.wikipedia.org/wiki/Hilbert_space#History where they go back to Hilbert and Schmidt studying integral equations, in particular eigenfunction expansions of the solutions. –  Gerald Edgar Feb 5 '12 at 17:53
See also this anecdote: mathoverflow.net/questions/53122/mathematical-urban-legends/… –  Todd Trimble Feb 5 '12 at 18:00
There is a long discussion of Hilbert's contributions in Dieudonne's History of Functional Analysis, although I don't feel knowledgeable enough to competently summarize it. –  Qiaochu Yuan Feb 5 '12 at 18:23
You forgot the most important theorem about Hilbert spaces: the spectral theorem. I believe Hilbert was the one who recognized that the work of Fredholm and others on integral operators could be strengthened by working over $L^2$ and generalizing what was then called the "principal axis theorem" to infinite dimensions. Indeed, he coined the term "spectrum" (long before the connection with physics), proved some version of the spectral theorem for self-adjoint operators, and discovered that the spectrum need not be discrete. He ought to get SOMETHING named after him for all that... –  Paul Siegel Feb 5 '12 at 18:26
I can only repeat what Qiaochu wrote: see Dieudonné's book, which has a full section called "The contributions of Hilbert", starting on p.105: to get a flavour, see amazon.fr/History-Functional-Analysis-Jean-Dieudonne/dp/… –  Alain Valette Feb 5 '12 at 20:46