English
A seminorm is convex on E with respect to the scalar field; more precisely, the map p is a convex functional on E over the reals.
Русский
Семинорм является выпуклой функциональной на E над действительными числами.
LaTeX
$$ConvexOn ℝ univ p$$
Lean4
/-- Seminorm-balls containing the origin are absorbent. -/
protected theorem absorbent_ball (hpr : p x < r) : Absorbent 𝕜 (ball p x r) :=
by
refine (p.absorbent_ball_zero <| sub_pos.2 hpr).mono fun y hy => ?_
rw [p.mem_ball_zero] at hy
exact p.mem_ball.2 ((map_sub_le_add p _ _).trans_lt <| add_lt_of_lt_sub_right hy)