a linear space endowed with a dot product and complete in the norm corresponding to that dot product
an inner product space that is complete with respect to the norm defined by the inner product (and is hence a Banach space )
an inner product space which, as a metric space, is complete
a particular kind of vector space with infinite dimensions
a vector space with an inner product such that the norm defined by turns into a complete metric space
a vector space over the complex numbers C with an inner product in which sequences that should converge actually do converge to points in the space. (for a more in depth definition)
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 )2 is defined as u · u as usual, and the space only contains those vectors whose length is finite.
In mathematics, a Hilbert space is a real or complex vector space with a positive-definite Hermitian form, that is complete under its norm. Thus it is an inner product space, which means that it has notions of distance and of angle (especially the notion of orthogonality or perpendicularity). The completeness requirement ensures that for infinite dimensional Hilbert spaces the limits exist when expected, which facilitates various definitions from calculus.
