Definition ball_norm (x : V) (eps : R) (y : V) := norm (minus y x) < eps.
Definition locally_norm (x : V) (P : V -> Prop) :=
exists eps : posreal, forall y, ball_norm x eps y -> P y.

Lemma is_filter_lim_locally_unique :
forall (x y : V),
is_filter_lim (locally x) y -> x = y.