Inner Product Space - Operators On Inner Product Spaces

Operators On Inner Product Spaces

Several types of linear maps A from an inner product space V to an inner product space W are of relevance:

• Continuous linear maps, i.e., A is linear and continuous with respect to the metric defined above, or equivalently, A is linear and the set of non-negative reals {||Ax||}, where x ranges over the closed unit ball of V, is bounded.
• Symmetric linear operators, i.e., A is linear and ⟨Ax, y⟩ = ⟨x, Ay⟩ for all x, y in V.
• Isometries, i.e., A is linear and ⟨Ax, Ay⟩ = ⟨x, y⟩ for all x, y in V, or equivalently, A is linear and ||Ax|| = ||x|| for all x in V. All isometries are injective. Isometries are morphisms between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with orthogonal matrix).
• Isometrical isomorphisms, i.e., A 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 operators on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.