---
tag: 0J3578562S
kind: definition
name: Equivalent norms
msc: [46B20]
defines: 0M85DXB5RM
uses: [0TWQ5PRKWS, 0A6YRNZB2R]
sources:
  - url: https://en.wikipedia.org/wiki/Norm_(mathematics)
    locator: Definition
  - url: https://en.wikipedia.org/wiki/Norm_(mathematics)
    locator: Equivalence
---

## Definition

Let $V$ be a vector space over $\mathbb{R}$ or $\mathbb{C}$.
A norm on $V$ is a function $\|\cdot\| \colon V \to [0, \infty)$ such that, for all $x, y \in V$ and all scalars $\lambda$: $\|x\| = 0$ only for $x = 0$, $\|\lambda x\| = |\lambda| \, \|x\|$, and $\|x + y\| \le \|x\| + \|y\|$.
Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $V$ are equivalent when there are real numbers $c > 0$ and $C > 0$ with
$$c \, \|x\|_a \le \|x\|_b \le C \, \|x\|_a \quad \text{for all } x \in V.$$

## Remarks

Each norm gives a metric $d(x, y) = \|x - y\|$. Two norms are equivalent if and only if their metrics are strongly [[equivalent|0TWQ5PRKWS]].
For norms, equivalence is the same as giving the same [[metric topology|0A6YRNZB2R]].
On a finite-dimensional space, [[all norms are equivalent|03MC70ZRNR]].
On $C([0, 1])$, the sup norm and the norm $\int_0^1 |f(t)| \, dt$ are not equivalent.

## Explanations

### L1 human

Equivalent norms measure the same vectors as small or large, up to a fixed factor.
So they agree on all questions of convergence and continuity.