# Norm is Continuous

Jump to navigation
Jump to search

## 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.

$\blacksquare$