Normed Division Ring Operations are Continuous/Addition

From ProofWiki
Jump to navigation Jump to search

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