The story is told by Saunders Mac Lane, quoted in the book Mathematical Apocrypha Redux: > J. von Neumann in 1927 introduced the axiomatic description of a Hil

The story is told by Saunders Mac Lane, quoted in the book Mathematical Apocrypha Redux: > J. von Neumann in 1927 introduced the axiomatic description of a Hilbert space, and used it in his work on quantum mechanics. There is a story of the time he came to Göttingen in 1929 to lecture on these ideas. The lecture started "A Hilbert space is a linear vector space over the complex numbers, complete in the convergence defined by an inner product (a product <a,b> of two vectors a, b) and separable". At the end of the lecture, David Hilbert (by custom sitting in the first row of the Mathematische Gesellschaft), who was then evidently thinking about his definition and not about the axiomatic description, is said to have asked, "Dr. von Neumann, ich möchte gern wissen, was ist dann eigentlich ein Hilbertscher Raum?" [Freely translated this is "Dr. von Neuman, I would very much like to know, what after all is a Hilbert space?"] https://books.google.com/books?id=8mBdvAjk_gQC&newbks=1&newbks_redir=0&lpg=PP1&dq=Mathematical%20Apocrypha%20Redux&pg=PA171#v=onepage&q&f=false https://books.google.com/books?id=8mBdvAjk_gQC&newbks=1&newb...