Theorems · Theorem · functional analysis
Summable.of_norm_bounded
∀ {ι : Type u_1} {E : Type u_3} [inst : SeminormedAddCommGroup E] [CompleteSpace E] {f : ι → E} {g : ι → ℝ},
Summable g → (∀ (i : ι), ‖f i‖ ≤ g i) → Summable fThe direct comparison test for series: if the norm of f is bounded by a real function g
which is summable, then f is summable.
- Cited by
- 25 results in Mathlib
- Foundations
- Depth 166 from the axioms · uses propext, Classical.choice, Quot.sound
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 and proof · cited by 25,697
- Norm.normstatement and proof · cited by 5,413
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- CompleteSpacestatement and proof · cited by 2,532
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- Summablestatement and proof · cited by 778
- summable_iff_cauchySeq_finsetproof · cited by 9
- cauchySeq_finset_of_norm_boundedproof · cited by 6
Cited by25
Results whose statement or proof uses this declaration.
- Summable.of_normproof · cited by 40
- summable_norm_iffproof · cited by 8
- HurwitzKernelBounds.summable_f_natproof · cited by 5
- hasFDerivAt_jacobiTheta₂proof · cited by 4
- summable_jacobiTheta₂_term_iffproof · cited by 4
- Summable.of_nnnorm_boundedproof · cited by 4
- PeriodPair.summable_weierstrassPExceptSummandproof · cited by 3
- tendsto_tsum_of_dominated_convergenceproof · cited by 2
- MeasureTheory.hasSum_integral_of_summable_integral_normproof · cited by 2
- summable_jacobiTheta₂'_term_iffproof · cited by 2
- summable_jacobiTheta₂_term_fderiv_iffproof · cited by 2
- HurwitzZeta.hasSum_int_completedCosZetaproof · cited by 2