Definition:Isometric Isomorphism

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Definition

Normed Division Ring

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.


Normed Vector Space

Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be normed vector spaces.

Let $T : X \to Y$ be a linear isometry.


We say that $T$ is an isometric isomorphism if and only if $T$ is bijective.


If an isometric isomorphism $T : X \to Y$ exists, we say that $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ are isometrically isomorphic.