Equivalent norms

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. For norms, equivalence is the same as giving the same metric topology. On a finite-dimensional space, all norms are equivalent. 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.