Completion of Normed Division Ring

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Theorem

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Then:

$\struct {R, \norm {\, \cdot \,} }$ has a normed division ring completion $\struct {R', \norm {\, \cdot \,}' }$


Proof

Let $d$ be the metric induced by $\struct {R, \norm {\, \cdot \,} }$.

Let $\mathcal C$ be the ring of Cauchy sequences over $R$.

Let $\mathcal N = \set {\sequence {x_n}: \displaystyle \lim_{n \mathop \to \infty} x_n = 0_R}$.

Let $\norm {\, \cdot \,}:\mathcal C \, \big / \mathcal N \to \R_{\ge 0}$ be the norm on the quotient ring $\mathcal C \, \big / \mathcal N$ defined by:

$\displaystyle \forall \sequence {x_n} + \mathcal N: \norm {\sequence {x_n} + \mathcal N } = \lim_{n \mathop \to \infty} \norm{x_n}$

Let $d'$ be the metric induced by $\struct {\mathcal C \, \big / \mathcal N, \norm {\, \cdot \,} }$.

By Quotient Ring of Cauchy Sequences is Normed Division Ring, $\struct {\mathcal C \, \big / \mathcal N, \norm {\, \cdot \,} }$ is a normed division ring.

By Quotient of Cauchy Sequences is Metric Completion, $\struct {\mathcal C \, \big / \mathcal N, d' }$ is the metric completion of $\struct {R, d}$.

Let $\phi: R \to \mathcal C \, \big / \mathcal N$ be the mapping from $R$ to the quotient ring $\mathcal C \,\big / \mathcal N$ defined by:

$\quad \quad \quad \forall a \in R: \map \phi a = \tuple {a, a, a, \ldots} + \mathcal N$

where $\tuple {a, a, a, \ldots} + \mathcal N$ is the left coset in $\mathcal C \, \big / \mathcal N$ that contains the constant sequence $\tuple {a, a, a, \ldots} $.

By Quotient of Cauchy Sequences is Metric Completion, $\map \phi R$ is a dense subset of $\struct {\mathcal C \, \big / \mathcal N, d' }$.

By Embedding Division Ring into Quotient Ring of Cauchy Sequences, $\phi$ is a distance-preserving ring monomorphism.

By the definition of a normed division ring completion, $\struct {\mathcal C \, \big / \mathcal N, \norm {\, \cdot \,} }$ is a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$.

$\blacksquare$


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