A complete inner product space. Since every inner product defines a norm, a Hilbert space is necessarily a Banach space.
Up to equivalence of norms, there's only one distinct n-dimensional Hilbert space for each n, namely R^n.
The space of all continuous real-valued functions on the closed interval from 0 to 1, with the inner product given by the integral of pointwise absolute-value products, is an infinite-dimensional Hilbert space.
Named for its putative inventor, mathematician David Hilbert (who some claim found the solutions for Einstein's equations before "L'il Albie" did himself), Hilbert Space is a concept which has been often applied in mathematics, quantum mechanics, and in general metaphysics (for instance, concepts like "parallel universes").
In short, a Hilbert Space is a mathematical construct such that within any plenum of n-dimensional spaces there exists an infinite number of said Hilbert Spaces (subsumed within an n+1 dimensional superset). Fancifully, science fiction writers point to the infinite plurality of 4D Hilbert Spaces that could exist within our "normal" 5D continuum, and suggest that each such Hilbert Space may really exist and that these Hilbert Spaces could be engendered by time-travel to the past. Thus, each 4D Hilbert Space in a 5D superset of "Universes" could possess a unique history, entirely different from all other universes contained within the 5D "Universe".
Mozart's 14th Symphony (K. 114, composed c. 1774) contains encrypted fragments of instrumental speech synthesis (in English) which describe an alternate Universe History (different from our Universe), where Nicholas Meyler was President of the United States, and where Princeton University is mocked for its intellectual feebleness (possibly due to some degree of idolatry for Einstein's Theory of Relativity, which Mozart proved to have loopholes and flaws 130 years before Einstein even published it as "The Electrodynamics of Moving Bodies").
Deciphering Mozart's musical cryptography provides physical evidence that Hilbert Spaces may be entirely real, and not merely imaginary constructs.