Suppose $V$ is a finite dimensional vector space. We know that all norms on $V$ are equivalent. Suppose the sequence $\{v_i\}$ converges to $v$ in some norm. Since convergence depends only on the topology, is it correct to say that the sequence converges to $v$ in all norms?


Yes, if you are working with the topology induced by the metrics​ induced by the respective norms. You can easily see this using limits, because the metrics are equivalent:

Let $\|\cdot\|\,,\;\|\cdot\|'$ denote equivalent norms on a vector space $V$. Assume $(x_n)_n$ has the limit $x$ w.r.t. the metric induced by $\|\cdot\|$. Due to norm equivalence, we have $c>0$ such that $0 \leq \|x - x_n\|' \leq c \|x - x_n\|$ for $n \in \mathbb{N}$...

Alternatively, if you want to think about it topologically, consider that norm equivalency implies that each open neighborhood.w.r.t. one norm-induced topology contains an open neighborhood w.r.t. the other norm-induced topology.
This follows by looking at how the open balls relate to each other due to norm equicalency. The argument then doesn't look much different from the above.


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