Definition:Isometric Isomorphism

Definition
Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings.

Let $d_R$ and $d_S$ be the metric induced by the norms $\norm {\,\cdot\,}_R$ and $\norm {\,\cdot\,}_S$ respectively.

Let $\phi:R \to S$ be a bijection such that:
 * $(1): \quad \phi: \struct {R, d_R} \to \struct {S, d_S}$ is an isometry.
 * $(2): \quad \phi: R \to S$ is a ring isomorphism.

Then $\phi$ is called an isometric isomorphism.

The normed division rings $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ are said to be isometrically isomorphic.