Section NVS_continuity.
Context {K : AbsRing} {V : NormedModule K}.
End NVS_continuity.

Lemma filterlim_locally_ball_norm :
forall {K : AbsRing} {T} {U : NormedModule K} {F : (T -> Prop) -> Prop} {FF : Filter F} (f : T -> U) (y : U),
filterlim f F (locally y) <-> forall eps : posreal, F (fun x => ball_norm y eps (f x)).