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 locally_norm_ball_norm :
forall (x : V) (eps : posreal),
locally_norm x (ball_norm x eps).