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,