Linear isometries and isometric isomorphisms

Let $V$ and $W$ be normed spaces over the same scalar field.

A linear map $T:V\to W$ (Linear map between vector spaces over the same field) is a linear isometry if $\mathrm{\parallel }Tx\mathrm{\parallel }=\mathrm{\parallel }x\mathrm{\parallel }\text{for every }x\in V\mathrm{.}$

A linear isometry is automatically an isometric embedding for the induced metrics, because $\mathrm{\parallel }T(x)-T(y)\mathrm{\parallel }=\mathrm{\parallel }T(x-y)\mathrm{\parallel }=\mathrm{\parallel }x-y\mathrm{\parallel }\mathrm{.}$

A linear isometric isomorphism is a linear isometry that is bijective (Injection, surjection, bijection). Equivalently, it is a bijective isometry between the underlying metric spaces whose map is linear.

• The adjective "isometric" by itself refers to the metric notion of Isometry, isometric embedding, and the subspace metric on a subset; the adjective "linear" is recorded separately because later pages need both embedding and surjective forms.
• A bijective linear isometry has a linear inverse, so a linear isometric isomorphism identifies two normed spaces without changing any distance.