Norm is Continuous

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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.

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples