Exercise 516Topology › Metric and normed spacesL1Added 3 Nov 2026

Normed spaces

#norm#definition#axioms

Statement

Define a norm on a vector space and show that the induced distance is a metric.

Hint

$\|x\|=0\iff x=0$ ; $\|\lambda x\|=|\lambda|\|x\|$ ; triangle inequality.

Show solutionHide solution

Axioms. A norm $\|\cdot\|:E\to[0,+\infty[$ satisfies:
1. $\|x\|=0\iff x=0$ ;
2. $\|\lambda x\|=|\lambda|\|x\|$ ;
3. $\|x+y\|\le\|x\|+\|y\|$ .

Induced metric. $d(x,y)=\|x-y\|$ is a metric: positivity from (1), symmetry from $\|\lambda x\|$ with $\lambda=-1$ , triangle from (3).

Conclusion. Every normed space is a metric space.