Hilbert space

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A Hilbert space is an inner product space which is complete with respect to the norm defined by the inner product (and is hence a Banach space). Hilbert spaces were named after David Hilbert, who studied them in the context of integral equations. Hilbert spaces serve to clarify and generalize the concept of Fourier expansion, certain linear transformations such as the Fourier transform, and are of crucial importance in the mathematical formulation of quantum mechanics. Hilbert spaces are studied in the branch of mathematics called functional analysis.

Examples of Hilbert spaces are Rn and Cn with the inner product definition <x, y> = ∑ xk yk*, where * denotes complex conjugation. Much more typical are the infinite dimensional Hilbert spaces however, in particular the space L2([a, b]) of square Lebesgue-integrable functions with values in R or C, modulo the subspace of those functions whose square integral is zero. The inner product of the two functions f and g is here given by

<f, g> = ∫ f(x) g(x)* dx

The use of the Lebesgue integral ensures that the space will be complete.

An important concept is that of an orthonormal basis of a Hilbert space H: a subset B of H with three properties:

  1. Every element of B has norm 1: <e, e> = 1 for all e in B
  2. Every two different elements of B are orthogonal: <e, f> = 0 for all e, f in B with ef.
  3. The span of B is dense in H.

Note that in the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter bases are also called Hamel-bases.

Using Zorn's lemma, one can show that every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality. A Hilbert space is separable if and only if it admits a countable orthonormal basis. If B is an orthonormal basis of H, then every element x of H may be written as

   x  =  ∑  <x,b> b

Even if B is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the Fourier expansion of x.

An important fact is that every Hilbert space is reflexive (see Banach space) and that one has a complete description of its dual space. Indeed, the Riesz representation theorem states that to every element φ of the dual H' there exists one and only one u in H such that

φ(x) = <x, u>     for all x in H

and the association φ <-> u provides an antilinear isomorphism between H and H'.

Need to mention operators, in particular self-adjoint and unitary

See also analysis, functional analysis, harmonic analysis.