The question Hilbert asked was an acute one of making this precise: is there any difference if a restriction to smooth manifolds is imposed?
Since the precise notion of "manifold" was not available to Hilbert, there is room for some debate about the formulation of the problem in contemporary mathematical language.
[3] The first major result was that of John von Neumann in 1933,[4] giving an affirmative answer for compact groups.
The final resolution, at least in the interpretation of what Hilbert meant given above, came with the work of Andrew Gleason, Deane Montgomery and Leo Zippin in the 1950s.
This was the subject of Per Enflo's doctoral thesis; his work is discussed in Benyamini & Lindenstrauss (2000, Chapter 17).