Theorems · Theorem · functional analysis
Summable.of_norm_bounded_eventually
∀ {ι : Type u_1} {E : Type u_3} [inst : SeminormedAddCommGroup E] [CompleteSpace E] {f : ι → E} {g : ι → ℝ},
Summable g → (∀ᶠ (i : ι) in Filter.cofinite, ‖f i‖ ≤ g i) → Summable fVariant of the direct comparison test for series: if the norm of f is eventually bounded by a
real function g which is summable, then f is summable.
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- Filter.Eventuallystatement and proof · cited by 3,134
- 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
- Filter.cofinitestatement and proof · cited by 251
- summable_iff_cauchySeq_finsetproof · cited by 9
- cauchySeq_finset_of_norm_bounded_eventuallyproof · cited by 2
Cited by13
Results whose statement or proof uses this declaration.
- summable_of_isBigOproof · cited by 8
- Summable.of_norm_bounded_eventually_natproof · cited by 6
- FormalMultilinearSeries.ofScalars_radius_eq_top_of_tendstoproof · cited by 3
- Summable.hasProdUniformlyOn_one_addproof · cited by 3
- FormalMultilinearSeries.inv_le_ofScalars_radius_of_tendstoproof · cited by 2
- summable_jacobiTheta₂'_term_iffproof · cited by 2
- summable_jacobiTheta₂_term_fderiv_iffproof · cited by 2
- Complex.summable_log_one_add_of_summableproof · cited by 2
- LSeriesSummable.congr'proof · cited by 2
- spectrum.exp_mem_expproof · cited by 1
- contDiff_tsum_of_eventuallyproof · cited by 1
- ZLattice.summable_norm_sub_rpowproof · cited by 1