Definition:Completion (Normed Division Ring)

Definition

Let $\struct {R_1, \norm {\, \cdot \,}_1}$ and $\struct {R_2, \norm {\, \cdot \,}_2}$ be normed division rings.

Let $M_1 = \struct {R_1, d_1}$ and $M_2 = \struct {R_2, d_2}$ be the metric spaces where $d_1: R_1 \times R_1 \to \R_{\ge 0}$ and $d_2: R_2 \times R_2 \to \R_{\ge 0}$ are the metrics induced by $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ respectively.

Then $\struct {R_2, \norm {\, \cdot \,}_2}$ is a completion of $\struct {R_1, \norm {\, \cdot \,}_1}$ if and only if:

$(1): \quad$ there exists a distance-preserving ring monomorphism $\phi: R_1 \to R_2$

$(2): \quad M_2$ is a metric completion of $\map \phi {M_1}$.

That is, $\struct {R_2, \norm{\,\cdot\,}_2}$ is a completion of $\struct {R_1,\norm{\,\cdot\,}_1}$ if and only if:

$(a): \quad M_2$ is a complete metric space

$(b): \quad$ there exists a distance-preserving ring monomorphism $\phi: R_1 \to R_2$

$(c): \quad \map \phi {R_1}$ is a dense subspace in $M_2$.

Also known as

$\struct {R_2, \norm {\, \cdot \,}_2}$ is a completion of $\struct {R_1, \norm {\, \cdot \,}_1}$

can also be worded as:

$\struct {R_2, \norm {\, \cdot \,}_2}$ completes $\struct {R_1, \norm {\, \cdot \,}_1}$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions
• 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.3$: Construction of the completion of a normed field