Isometric isomorphism

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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

literature