linear algebra, higher linear algebra
(…)
A linear isometry is a linear map
between normed vector spaces which preserves the norm:
In particular, when the norm is induced by (Hermitian) inner products ${\langle \cdot \vert \cdot \rangle}_{i}$ (as on Hilbert spaces)
then a linear isometry is an isometry in that it preserves these inner products:
or equivalently, in terms of adjoint operators:
(relation to unitary operators)
If in addition to (1) also the reverse condition $\phi \cdot \phi^\dagger \,=\, Id_{\mathscr{H}_2}$ holds, then $\phi$ is called a unitary operator, which is the case iff $\phi$ is surjective map.
Linear isometries are injective maps.
For $\left\vert \psi \right\rangle, \left\vert \psi' \right\rangle \,\colon\, \mathscr{H}$ we need to show that
But by non-degeneracy of the norm this is equivalent to showing
But since $\phi$ is an isometry, here the left hand side already coincides with the right hand side and hence certainly implies it.
See also:
ProofWiki, Linear Isometry
Wikipedia, Linear isometry
Last revised on September 26, 2023 at 06:03:04. See the history of this page for a list of all contributions to it.