End Riemann_sum.
Section Riemann_sum_Normed.
Context {V : NormedModule R_AbsRing}.

Lemma Riemann_sum_norm (f : R -> V) (g : R -> R) ptd :
pointed_subdiv ptd ->
(forall t, SF_h ptd <= t <= last (SF_h ptd) (SF_lx ptd) -> norm (f t) <= g t)
-> norm (Riemann_sum f ptd) <= Riemann_sum g ptd.