Isometric isomorphism
In functional analysis, isometric isomorphism describes a relationship between two different spaces that are geometrically identical.
definition
Two normalized spaces and are isometrically isomorphic if a vector space isomorphism exists between them , which is at the same time an isometry , i.e. fulfills. Then you write .
This means that the rooms can be clearly identified with one another and length measurements can be transferred from one to the other. The operator takes over the identification of elements from with elements from The isometry of ensures that the standard is maintained with this change. Apparently the inversion is again an isometric isomorphism.
Examples
- Every separable infinite-dimensional Hilbert space is isometrically isomorphic to the space of all sequences with the property that the sum of the squares of all sequence terms is finite.
- Two Hilbert spaces are isometrically isomorphic if and only if their Hilbert space dimensions match.
- Every normalized vector space is isometrically isomorphic to a sub-vector space of the space of continuous functions on a suitably chosen compact topological space according to the supreme norm .
- According to the Banach-Mazur theorem , every separable, normed space is isometrically isomorphic to a subspace of the space of continuous functions from the unit interval to with the supremum norm .
literature
- Dirk Werner : Functional Analysis . Springer Verlag, 2005. ISBN 3-540-43586-7