Hilbert space

John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications.

At a deeper level, perpendicular projection onto a linear subspace plays a significant role in optimization problems and other aspects of the theory.

[18] John von Neumann coined the term abstract Hilbert space in his work on unbounded Hermitian operators.

[19] Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from a physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them.

The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups.

[22] The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics.

[26] In the 1940s, Israel Gelfand, Mark Naimark and Irving Segal gave a definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied.

Furthermore, another consequence of this general result is that the eigenvalues λ of the system can be arranged in an increasing sequence tending to infinity.

For linear elliptic equations, one geometrical result that ensures unique solvability for a large class of problems is the Lax–Milgram theorem.

[41] Hilbert spaces allow for many elliptic partial differential equations to be formulated in a similar way, and the Lax–Milgram theorem is then a basic tool in their analysis.

The protypical case of a field that ergodic theory applies to is thermodynamics, in which—though the microscopic state of a system is extremely complicated (it is impossible to understand the ensemble of individual collisions between particles of matter)—the average behavior over sufficiently long time intervals is tractable.

This isometry property of the Fourier transformation is a recurring theme in abstract harmonic analysis (since it reflects the conservation of energy for the continuous Fourier Transform), as evidenced for instance by the Plancherel theorem for spherical functions occurring in noncommutative harmonic analysis.

The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.

The Itô integral can be constructed by first defining it for simple processes, and then exploiting their density in the Hilbert space.

A deeper application of Hilbert spaces that is especially important in the theory of Gaussian processes is an attempt, due to Leonard Gross and others, to make sense of certain formal integrals over infinite dimensional spaces like the Feynman path integral from quantum field theory.

Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's theorem).

The (geometrical) Hahn–Banach theorem asserts that a closed convex set can be separated from any point outside it by means of a hyperplane of the Hilbert space.

This is an immediate consequence of the best approximation property: if y is the element of a closed convex set F closest to x, then the separating hyperplane is the plane perpendicular to the segment xy passing through its midpoint.

Examples of self-adjoint unbounded operators on the Hilbert space L2(R) are:[85] These correspond to the momentum and position observables, respectively.

The Hilbertian tensor product of two copies of L2([0, 1]) is isometrically and linearly isomorphic to the space L2([0, 1]2) of square-integrable functions on the square [0, 1]2.

This mapping defined on simple tensors extends to a linear identification between H1 ⊗ H2 and the space of finite rank operators from H∗1 to H2.

Geometrically, Bessel's inequality implies that the orthogonal projection of x onto the linear subspace spanned by the fi has norm that does not exceed that of x.

However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number of degrees of freedom and any infinite Hilbert tensor product (of spaces of dimension greater than one) is non-separable.

[98] For instance, a bosonic field can be naturally thought of as an element of a tensor product whose factors represent harmonic oscillators at each point of space.

[98] However, it is only a small separable subspace of the full tensor product that can contain physically meaningful fields (on which the observables can be defined).

The set S⊥ is a closed subspace of H (can be proved easily using the linearity and continuity of the inner product) and so forms itself a Hilbert space.

Unlike with finite matrices, not every element of the spectrum of T must be an eigenvalue: the linear operator T − λ may only lack an inverse because it is not surjective.

The general spectral theorem for self-adjoint operators involves a kind of operator-valued Riemann–Stieltjes integral, rather than an infinite summation.

A somewhat similar spectral decomposition holds for normal operators, although because the spectrum may now contain non-real complex numbers, the operator-valued Stieltjes measure dEλ must instead be replaced by a resolution of the identity.

In Gravity's Rainbow (1973), a novel by Thomas Pynchon, one of the characters is called "Sammy Hilbert-Spaess", a pun on "Hilbert Space".