[Def]:

Let $V$ be a vector space, $\|\cdot\|{N1}$ and $\|\cdot\|{N2}$ be two norms defined on $V$.

If $\exists \alpha,\beta>0$ s.t. $\alpha\|v\|{N1} \leq \|v\|{N2} \leq \beta\|v\|_{N1}$, for all $v\in V$.

Then we say $\|\cdot\|{N1}\sim \|\cdot\|{N2}$

[Note]: If $\|\cdot\|{N1}\sim \|\cdot\|{N2}$, then $\|x_n-x_\ast\|{N1}\xrightarrow[n\rightarrow\infty]{}0$ iff $\|x_n-x\ast\|_{N2}\xrightarrow[n\rightarrow\infty]{}0$