Hilbert Spaces

Hilbert Spaces and Separability

Hilbert Space

A Hilbert space is a real or complex inner product space that is also complete with respect to the metric induced by the inner product.

Separable Nomed Space

A normed space is separable if it has a countable dense subset. i.e. it contains a countable subset whose closure is the entire space.

e.g.

• The finite dimensional complex spaces are separable Hilbert spaces, with the inner product being
• An infinite-dimensional analogue of the above example is the space with the inner product
• The space of all complex-valued measurable function on such that , modulo the equivalence relation of being equal almost everywhere, denoted as , is a prime example of a separable Hilbert space. The inner product is defined as We shall first check that is a vector space, and is a well-defined inner product. For any , we have so for all , hence is a vector space. Moreover, So the inner product is well-defined. Now we verify that is complete.
• Indeed, is a Hilbert space if and only if . For , is only a Banach space.

Orthonormal Basis

Bessel's Inequality

Let be a Hilbert space, and let ​ be an orthonormal set in . For any , there holds

Characterisation of Orthonormal Basis in Hilbert Space

The following properties of a countable orthonormal set in a Hilbert space are equivalent:

• forms a basis.
• The span of is dense in .
• If and for all , then .
• If , and , then as .
• For all , . This is called the Parseval’s identity

Proof The Parseval’s identity is a direct result of the Pythagorean theorem.

Theorem

A Hilbert space has an orthonormal basis if and only if it is separable.

Proof Suppose is a separable Hilbert space and is a countable subset of whose closure equals . We will inductively define an orthonormal basis such that for all . This will imply that , which will mean that is an orthonormal basis. Without loss of generality, assume that . First set . Then suppose for some , is an orthonormal set such that . If for every , then is an orthonormal basis of and the process should be stopped. Otherwise, let be the smallest positive integer such that Define as follows: Clearly , for all , our choice of guarantees there is no division by , and completing the induction.

Same Gram-Schmidt Procedure in Linear Algebra

This is actually the same Gram-Schmidt procedure in linear algebra.

e.g. Consider the space , the set is an orthonormal basis, and for each function , the corresponding th coefficient is the th Fourier coefficient: This can be seen by the following two facts:

• The Parseval’s identity holds for :
• The Fourier series converges to in the norm:

Indeed, the mapping makes (unitarily) “equivalent” to , we shall seriously define this equivalence, and prove this in the next section. (See the proposition.)

Unitary Mappings

Unitary Mapping

A mapping between two Hilbert spaces is called unitary if it is a vector space isomorphism and preserves the inner product, i.e. for all . We call two Hilbert spaces unitarily equivalent if there exists a unitary mapping between them.

Theorem

Any two separable Hilbert spaces are unitarily equivalent if and only if they have the same dimension.

Proof

Proposition

is unitarily equivalent to under the mapping , where .

Proof

Pre-Hilbert Spaces

Pre-Hilbert Space

A Pre-Hilbert space is a real or complex inner product space that is not necessarily complete.

Every Pre-Hilbert Space Can be Completed

Suppose we are given a pre-Hilbert space . Then there exists a unique (up to unitarily equivalence) Hilbert space , such that is dense, and is the restriction of to .

Proof

Fatou’s Theorem

Fatou's Theorem

A bounded holomorphic function on the unit disk has radial limits at almost every .