Normed Division Ring Operations are Continuous/Addition
Theorem
Let $\struct {R, +, *, \norm {\,\cdot\,} }$ be a normed division ring.
Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.
Let $p \in \R_{\ge 1} \cup \set \infty$.
Let $d_p$ be the $p$-product metric on $R \times R$.
Then the mapping:
- $+ : \struct {R \times R, d_p} \to \struct{R,d}$
is continuous.
Proof
By $p$-Product Metric Induces Product Topology and Continuous Mapping is Continuous on Induced Topological Spaces, it suffices to consider the case $p = \infty$.
Let $\tuple {x_0, y_0} \in R \times R$.
Let $\epsilon > 0$ be given.
Let $\tuple {x, y} \in R \times R$ such that:
- $\map {d_\infty} {\tuple {x, y}, \tuple{x_0, y_0} } < \dfrac \epsilon 2$
By the definition of the product metric $d_\infty$ then:
- $\max \set {\map d {x, x_0}, \map d {y, y_0} } < \dfrac \epsilon 2$
or equivalently:
- $\map d {x, x_0} < \dfrac \epsilon 2$
- $\map d {y, y_0} < \dfrac \epsilon 2$
Then:
\(\ds \map d {x_0 + y_0, x + y}\) | \(=\) | \(\ds \norm {\paren {x_0 + y_0} - \paren {x + y} }\) | Definition of Metric Induced by Norm on Division Ring | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm {\paren {x_0 - x} + \paren {y_0 - y} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \norm {x_0 - x } + \norm {y_0 - y}\) | Norm Axiom $\text N 3$: Triangle Inequality | |||||||||||
\(\ds \) | \(\le\) | \(\ds \map d {x_0, x} + \map d {y_0, y}\) | Definition of Metric Induced by Norm on Division Ring | |||||||||||
\(\ds \) | \(<\) | \(\ds \dfrac \epsilon 2 + \dfrac \epsilon 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \epsilon\) |
Since $\tuple {x_0, y_0}$ and $\epsilon$ were arbitrary, by the definition of continuity then the mapping:
- $+ : \struct {R \times R, d_\infty} \to \struct {R, d}$
is continuous.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.3$: Topology, Problem $43$