Editing Inner product space (section)

==Operators on inner product spaces==
{{Main|Operator theory}}
Several types of [[linear]] maps <math>A : V \to W</math> between inner product spaces <math>V</math> and <math>W</math> are of relevance:
* {{em|[[Continuous linear operator|Continuous linear maps]]}}: <math>A : V \to W</math> is linear and continuous with respect to the metric defined above, or equivalently, <math>A</math> is linear and the set of non-negative reals <math>\{ \|Ax\| : \|x\| \leq 1\},</math> where <math>x</math> ranges over the closed unit ball of <math>V,</math> is bounded.
* {{em|Symmetric linear operators}}: <math>A : V \to W</math> is linear and <math>\langle Ax, y \rangle = \langle x, Ay \rangle</math> for all <math>x, y \in V.</math>
* {{em|[[Isometry|Isometries]]}}: <math>A : V \to W</math> satisfies <math>\|A x\| = \|x\|</math> for all <math>x \in V.</math> A {{em|linear isometry}} (resp. an {{em|[[Antilinear map|antilinear]] isometry}}) is an isometry that is also a linear map (resp. an [[antilinear map]]). For inner product spaces, the [[polarization identity]] can be used to show that <math>A</math> is an isometry if and only if <math>\langle Ax, Ay \rangle = \langle x, y \rangle</math> for all <math>x, y \in V.</math>  All isometries are [[injective]]. The [[Mazur–Ulam theorem]] establishes that every surjective isometry between two {{em|real}} normed spaces is an [[affine transformation]]. Consequently, an isometry <math>A</math> between real inner product spaces is a linear map if and only if <math>A(0) = 0.</math> Isometries are [[morphism]]s between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with [[orthogonal matrix]]). 
* {{em|Isometrical isomorphisms}}: <math>A : V \to W</math> is an isometry which is [[surjective]] (and hence [[bijective]]). Isometrical isomorphisms are also known as unitary operators (compare with [[unitary matrix]]).

From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The [[spectral theorem]] provides a canonical form for symmetric, unitary and more generally [[normal operator]]s on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.<ref>{{harvnb|Rudin|1991}}</ref>