Theorems · Theorem · functional analysis
Summable.of_norm_bounded_eventually_nat
∀ {E : Type u_3} [inst : SeminormedAddCommGroup E] [CompleteSpace E] {f : ℕ → E} {g : ℕ → ℝ},
Summable g → (∀ᶠ (i : ℕ) in Filter.atTop, ‖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
- 6 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.
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
- Filter.atTopstatement and proof · cited by 2,405
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- Summablestatement and proof · cited by 778
- Nat.cofinite_eq_atTopproof · cited by 37
- Summable.of_norm_bounded_eventuallyproof · cited by 13
Cited by6
Results whose statement or proof uses this declaration.
- HurwitzKernelBounds.summable_f_natproof · cited by 5
- EisensteinSeries.summable_one_div_norm_rpowproof · cited by 4
- LSeriesSummable_of_le_const_mul_rpowproof · cited by 3
- summable_of_ratio_norm_eventually_leproof · cited by 2
- DirichletCharacter.eulerProduct_log_eq_LSeriesproof · cited by 1
- LSeries.abscissaOfAbsConv_logMulproof · cited by 1