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:

  • The inner product of a pair of elements is equal to the complex conjugate of the inner product of the swapped elements:
  • The inner product is linear in its first argument. For all complex numbers a and b,
  • The inner product of an element with itself is positive definite:
where the case of equality holds precisely when x = 0.

It follows from properties 1 and 2 that a complex inner product is antilinear in its second argument, meaning that

A real inner product space is defined in the same way, except that H is a real vector space and the inner product takes real values. Such an inner product will be bilinear: that is, linear in each argument.

The norm is the real-valued function

and the distance d between two points x,y in H is defined in terms of the norm by

That this function is a distance function means (1) that it is symmetric in x and y, (2) that the distance between x and itself is zero, and otherwise the distance between x and y must be positive, and (3) that the triangle inequality holds, meaning that the length of one leg of a triangle xyz cannot exceed the sum of the lengths of the other two legs:

This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality, which asserts

with equality if and only if x and y are linearly dependent.

Relative to a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a pre-Hilbert space. Any pre-Hilbert space that is additionally also a complete space is a Hilbert space. Completeness is expressed using a form of the Cauchy criterion for sequences in H: a pre-Hilbert space H is complete if every Cauchy sequence converges with respect to this norm to an element in the space. Completeness can be characterized by the following equivalent condition: if a series of vectors converges absolutely in the sense that

then the series converges in H, in the sense that the partial sums converge to an element of H.

As a complete normed space, Hilbert spaces are by definition also Banach spaces. As such they are topological vector spaces, in which topological notions like the openness and closedness of subsets are well-defined. Of special importance is the notion of a closed linear subspace of a Hilbert space that, with the inner product induced by restriction, is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right.

Read more about Hilbert Space:  History, Applications, Orthonormal Bases, Orthogonal Complements and Projections, Spectral Theory

Other articles related to "space, hilbert space, hilbert":

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 g (X) are a (closed, linear) subspace ... approximations to the infinite-dimensional Hilbert space ... = P ( Y ≤ 1/3 ), namely, However, the Hilbert space approach treats g2 as an equivalence class of functions rather than an individual function ...
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 ... 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 by demanding additional structure on the ... linearly independent projector operators so as to form a basis for the Hilbert-Schmidt space ...
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 ...
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 ... 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 ...
Hilbert Space - Spectral Theory
... There is 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 ... integral operators are compact, in particular those that arise from Hilbert–Schmidt operators ... the spectral theorem in this case is Given a densely defined self-adjoint operator T on a Hilbert space H, there corresponds a unique resolution of the identity E on the ...

Famous quotes containing the word space:

    Here were poor streets where faded gentility essayed with scanty space and shipwrecked means to make its last feeble stand, but tax-gatherer and creditor came there as elsewhere, and the poverty that yet faintly struggled was hardly less squalid and manifest than that which had long ago submitted and given up the game.
    Charles Dickens (1812–1870)