Normed Division Ring Operations are Continuous/Multiplication
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 $\delta = \min \set {\dfrac \epsilon {1 + \norm {y_0} + \norm {x_0} }, 1}$
Since $1 + \norm {y_0} + \norm {x_0} > 0$ then $\delta > 0$
Let $\tuple {x,y} \in R \times R$ such that:
- $\map {d_\infty} {\tuple {x, y}, \tuple{x_0, y_0} } < \delta$
By the definition of the $p$-product metric $d_\infty$:
- $\max \set {\map d {x, x_0}, \map d {y, y_0}} < \delta$
or equivalently:
- $\map d {x, x_0} < \delta$
- $\map d {y, y_0} < \delta$
Then:
\(\ds \norm y\) | \(=\) | \(\ds \norm {y - y_0 + y_0}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \norm {y - y_0} + \norm {y_0}\) | Norm Axiom $\text N 3$: Triangle Inequality | |||||||||||
\(\ds \) | \(\le\) | \(\ds \map d {y, y_0} + \norm {y_0}\) | Definition of Metric Induced by Norm on Division Ring | |||||||||||
\(\ds \) | \(<\) | \(\ds \delta + \norm {y_0}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds 1 + \norm {y_0}\) |
Hence:
\(\ds \map d {x y, x_0 y_0}\) | \(=\) | \(\ds \norm {x y - x_0 y_0}\) | Definition of Metric Induced by Norm on Division Ring | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm {x y - x_0 y + x_0 y - x_0 y_0}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \norm {x y - x_0 y} + \norm {x_0 y - x_0 y_0}\) | Norm Axiom $\text N 3$: Triangle Inequality | |||||||||||
\(\ds \) | \(\le\) | \(\ds \norm {x - x_0} \norm y + \norm {x_0} \norm {y - y_0}\) | Norm Axiom $\text N 2$: Multiplicativity | |||||||||||
\(\ds \) | \(\le\) | \(\ds \map d {x, x_0} \norm y + \norm {x_0} \map d {y, y_0}\) | Definition of Metric Induced by Norm on Division Ring | |||||||||||
\(\ds \) | \(<\) | \(\ds \delta \norm y + \norm {x_0} \delta\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \delta \paren {\norm y + \norm {x_0} }\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \delta \paren {1 + \norm {y_0} + \norm {x_0} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \dfrac \epsilon {1 + \norm {y_0} + \norm {x_0} } \paren {1 + \norm {y_0} + \norm {x_0} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \epsilon\) |
We have that $\tuple {x_0, y_0}$ and $\epsilon$ are arbitrary.
Hence, by the definition of continuity, 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 $43$