Norm is Continuous
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 It has been suggested that this page or section be merged into Norm on Vector Space is Continuous Function. (Discuss) |
Theorem
Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space.
Then the mapping $x \mapsto \norm x$ is continuous.
Here, the metric used is the metric $d$ induced by $\norm {\,\cdot\,}$.
Proof
Since $\norm x = \map d {x, \mathbf 0}$, the result follows directly from Distance Function of Metric Space is Continuous.
$\blacksquare$