Theorems · Theorem · sequences and series
multipliable_one_add_of_summable
∀ {ι : Type u_1} {R : Type u_2} [inst : NormedCommRing R] [NormOneClass R] {f : ι → R} [CompleteSpace R],
(Summable fun i => ‖f i‖) → Multipliable fun i => 1 + f iIn a complete normed ring, ∏' i, (1 + f i) is convergent if the sum of real numbers
∑' i, ‖f i‖ is convergent.
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 280 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites54
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- Realstatement and proof · cited by 25,697
- Finsetproof · cited by 13,712
- Norm.normstatement and proof · cited by 5,413
- mul_oneproof · cited by 3,885
- LE.le.transproof · cited by 3,151
- one_mulproof · cited by 2,841
- CompleteSpacestatement and proof · cited by 2,532
- Nat.cast_oneproof · cited by 2,501
- Filter.atTopproof · cited by 2,405
- Finset.prodproof · cited by 2,356
- Disjointproof · cited by 2,201
Cited by4
Results whose statement or proof uses this declaration.
- Summable.hasProdUniformlyOn_one_addproof · cited by 3
- tprod_one_add_ne_zero_of_summableproof · cited by 1
- multipliable_sineTermproof · cited by 1
- ModularForm.multipliable_one_sub_powproof · cited by 1