Theorems · Theorem · functional analysis
norm_prod_le_of_le
∀ {ι : Type u_3} {E : Type u_5} [inst : SeminormedCommGroup E] (s : Finset ι) {f : ι → E} {n : ι → ℝ},
(∀ b ∈ s, ‖f b‖ ≤ n b) → ‖∏ b ∈ s, f b‖ ≤ ∑ b ∈ s, n b- Defined in
- Mathlib.Analysis.Normed.Group.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SeminormedCommGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Finsetstatement and proof · cited by 13,712
- Norm.normstatement and proof · cited by 5,413
- Finset.sumstatement · cited by 5,195
- LE.le.transproof · cited by 3,151
- Finset.prodstatement · cited by 2,356
- SeminormedCommGroupstatement and proof · cited by 191
- Finset.sum_le_sumproof · cited by 182
- norm_prod_leproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- dist_prod_prod_le_of_leproof · cited by 1
- nnnorm_prod_le_of_leproof · cited by 0