Definition:Isometric Isomorphism/Normed Division Ring

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.

Also see

• Results about isometric isomorphisms on normed division rings can be found here.