# What is Hilbert space?

• (noun): A metric space that is linear and complete and (usually) infinite-dimensional.

## Hilbert Space

A Hilbert space H is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product. To say that H is a complex inner product space means that H is a complex vector space on which there is an inner product associating a complex number to each pair of elements x,y of H that satisfies the following properties:

### Some articles on Hilbert space:

Hilbert Space - Spectral Theory
... a well-developed spectral theory for self-adjoint operators in a Hilbert space, that is roughly analogous to the study of symmetric matrices over the reals or self ... as many integral operators are compact, in particular those that arise from Hilbert–Schmidt operators ... defined self-adjoint operator T on a Hilbert space H, there corresponds a unique resolution of the identity E on the Borel sets of R, such that for all x ∈ D(T) and y ∈ H ...
Conditioning (probability) - Conditioning On The Level of Measure Theory - Conditional Probability
... Indeed, the space L2 (Ω) of all square integrable random variables is a Hilbert space the indicator I is a vector of this space and random variables of the form ... numerically, using finite-dimensional approximations to the infinite-dimensional Hilbert space ... (unconditional) probability, E ( P ( Y ≤ 1/3
SIC-POVM - Definition
... In general, a POVM over a finite d-dimensional Hilbert space is defined as a set of positive semidefinite operators on a Hilbert space H that sum to unity, While a SIC-POVM will still ... if is a rank one projector in a d-dimensional Hilbert space, then the corresponding subnormalized projector is Furthermore, SIC-POVMs add to the theory of general POVMs ... must consist of linearly independent projector operators so as to form a basis for the Hilbert-Schmidt space ...
Constraint Algebra
... In theoretical physics, a constraint algebra is a linear space of all constraints and all of their polynomial functions or functionals whose action on the physical vectors of the Hilbert ... In quantum electrodynamics, one first constructs a Hilbert space in which Gauss' law does not hold automatically ... The true Hilbert space of physical states is constructed as a subspace of the original Hilbert space of vectors that satisfy In more general theories, the constraint algebra may be a ...
Commutation Theorem - Hilbert Algebras - Properties
... Let H be the Hilbert space completion of with respect to the inner product and let J denote the extension of the involution to a conjugate-linear involution of H ... In this case the commutation theorem for Hilbert algebras states that Moreover if the von Neumann algebra generated by the operators λ(a), then These results were proved independently by Godement (1954 ... the notion of "bounded elements" in the Hilbert space completion H ...

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