Theorems · Theorem · functional analysis
summable_mul_of_summable_norm
∀ {R : Type u_1} {ι : Type u_2} {ι' : Type u_3} [inst : NormedRing R] [CompleteSpace R] {f : ι → R} {g : ι' → R},
(Summable fun x => ‖f x‖) → (Summable fun x => ‖g x‖) → Summable fun x => f x.1 * g x.2- Defined in
- Mathlib.Analysis.Normed.Ring.InfiniteSum
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NormedRingCompleteSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- Norm.normstatement and proof · cited by 5,413
- CompleteSpacestatement and proof · cited by 2,532
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- NormedRingstatement and proof · cited by 924
- Summablestatement and proof · cited by 778
- Summable.of_normproof · cited by 40
- Summable.mul_normproof · cited by 2
Cited by5
Results whose statement or proof uses this declaration.
- LSeriesHasSum.convolutionproof · cited by 3
- EulerProduct.summable_and_hasSum_factoredNumbers_prod_filter_prime_tsumproof · cited by 3
- tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_normproof · cited by 2
- tsum_eisSummand_eq_riemannZeta_mul_eisensteinSeriesproof · cited by 1
- tsum_mul_tsum_of_summable_normproof · cited by 1