For a vector topological space, a sequence $(x_n)$ is Cauchy when $\forall$ neighbourhood $U$ of $0$, there exists $N$ such that $m,n>N$ implies $x_m-x_n\in U$.
For a metric topological space $(E,d)$, a sequence $(x_n)$ is Cauchy when $\forall \epsilon>0$, there exists $N$ such that $m,n>N$ implies $d(x_m,x_n)<\epsilon$.
QUESTION : are these two definitions equivalent in the case of a metric topological vector space ?
If the distance is invariant by translation, I can prove it. But in general no.
QUESTION : if the previous question has answer "no", what is the "good" definition in the intersection ?