1) Quantum mechanic 
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possess...
in mathematics, an example of an infinite-dimensional space that had a major impact in analysis and topology. The German mathematician David Hilbert ... [7 related articles]
A space of infinite dimensions, named after David Hilbert, in which distance is preserved by making the sum of squares of coordinates a convergent sequence; it is of crucial importance in the mathematical formulation of quantum mechanics.
A complete normed metric space with an inner product. So the Hilbert spaces are also Banach spaces. L2 is an example of a Hilbert space. Any Rn with n finite is another. Source: Royden p. 245 Contexts: real analysis
Vector space with infinitely many dimensions. The dot product u · v = Sum ui vi (or Sum ui vi* for spaces on the complex numbers) is replaced by the corresponding infinite sum or integral (Hilbert spaces defined either way are identical). The (length of u)2 is defined as u · u as usual, and the space only contains those vectors whose length is finite.
...
[n] - a metric space that is linear and complete and (usually) infinite-dimensional
noun a metric space that is linear and complete and (usually) infinite-dimensional
a complete infinite-dimensional vector space on which an inner product is defined.