Let $V$ be a [vector space](/page/Vector%20Space) over $\mathbb{F}$, where $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$. Let $\|\cdot\|_a:V\to[0,\infty)$ and $\|\cdot\|_b:V\to[0,\infty)$ be norms on $V$. Suppose the two norms are equivalent, meaning that there exist constants $m,M>0$ such that

\begin{align*} m\|v\|_a\leq \|v\|_b\leq M\|v\|_a \end{align*}

for every $v\in V$. Then the topology induced by $\|\cdot\|_a$ is equal to the topology induced by $\|\cdot\|_b$. Moreover, for every sequence $(x_n)_{n=1}^{\infty}$ in $V$ and every $x\in V$, one has $x_n\to x$ with respect to $\|\cdot\|_a$ if and only if $x_n\to x$ with respect to $\|\cdot\|_b$. Finally, for every subset $S\subset V$, the set $S$ is bounded with respect to $\|\cdot\|_a$ if and only if $S$ is bounded with respect to $\|\cdot\|_b$.