Importance of equivalence of norms

Question

I am doing a course in Convex Optimization where I learned about the equivalence of norms, but there was no mention of its importance or about the scenarios where it can come in handy. The definition I am referring to here is as follows.

Two norms $||x||_a$ and $||x||_b$ are said to be equivalent if $\exists \alpha, \beta$ s.t $\alpha$, $\beta$ > 0

$\alpha ||x||_b \leq ||x||_a \leq \beta ||x||_b$

Answer 1

On $\mathbb R^{m}$, all norms are equivalent in the sense that if $\| \|_{a}$ and $\| \|_{b}$ are norms, then there are constants $C_{1}$ and $C_{2}$ such that for every $x$ in $\mathbb R^{m}$,

$C_{1} \| x \|_{a} \leq \| x \|_{b} \leq C_{2} \| x \|_{a}$

An easy consequence of this is that if $x^{(n)}$ is a sequence of vectors such that

$\lim_{n \rightarrow \infty} \| x^{(n)} \|_{a}=0$

then

$\lim_{n \rightarrow \infty} \| x^{(n)} \|_{b}=0$.

This means that if you want to prove that $\| x^{(n)} \|_{a}$ converges to 0, you can pick any convenient norm and use it in the proof. For example, if you want to show that

$\lim_{n \rightarrow \infty} \| x^{(n)} \|_{2}=0$,

you can start by showing that for each component $x_{i}$,

$\lim_{n \rightarrow \infty} | x^{n}_{i} | = 0$,

then

$\lim_{n \rightarrow \infty} \sum_{i=1}^{m} | x^{(n)}_{i} | = 0$

or

$\lim_{n \rightarrow \infty} \| x^{(n)} \|_{1}=0$

and from there, you can use the equivalence of norms to conclude that

$\lim_{n \rightarrow \infty} \| x^{(n)} \|_{2}=0$.

Because of the equivalence of norms in $\mathbb R^{m}$, we say that

$\lim_{n \rightarrow \infty} x^{(n)}=x^{*}$

if there is some norm in which

$\lim_{n \rightarrow \infty} \| x^{(n)}-x^{*} \|=0$.

It simply doesn't matter what norm is used.

Answer 2

It's not hard to prove that norm equivalence is an equivalence relation. As Brian wrote, two equivalent norms have the same cauchy and convergent sequences, and therefore convergence/cauchyness of a sequence is a property of an equivalence class of norms.

All norms for a finite dimensional vector space (over $\mathbb{R}$ or $\mathbb{C}$) are equivalent (proof), as in the case of $\mathbb{R}^m$ mentioned by Brian. Consequently we can talk about about convergence in finite dimensional normed vector spaces without any need to specify the norm. What's more, is that equivalent norms generate the same standard topology (all sets which are unions of open balls in the given norm). Therefore we can talk about the standard normed vector space topology without needing to specify a norm (again this is just in finite dimensions).

Some good exercises are here, which might indicate some neat facts about norms.