In mathematics, particularly in the branch known as functional analysis, a Hilbert space is a complete inner product space. As such, it is automatically also a Banach space. Due to the presence of an inner product, a Hilbert space has additional useful geometric properties that are not found in a Banach space; these properties may be exploited to simplify analysis or to obtain stronger results. For example, in a Hilbert space there exist projection operators onto closed subpaces of the Hilbert space. This makes Hilbert spaces extremely important in certain branches of mathematics such as optimization and approximation theory. As another example, any complex (real) continuous linear functional over a complex (real) Hilbert space can be expressed as the inner product of elements of that space with some fixed element of the space (this is known as the Riesz representation theorem). In mathematical terms, one says that a Hilbert space is isomorphic to its dual and is actually its own dual if the Hilbert space is real.
In physics, Hilbert spaces play a fundamental role in the physical theory of quantum mechanics. The state (in the so-called Schrödinger picture) of a quantum mechanical system is postulated to be a unit vector (i.e., a vector of norm 1) in some Hilbert space, and physical quantities or "observables" are postulated as self-adjoint operators on that Hilbert space. States serve to assign statistical properties to observables of the system.