Theorems · Theorem · functional analysis
cauchySeq_finset_iff_vanishing_norm
∀ {ι : Type u_1} {E : Type u_3} [inst : SeminormedAddCommGroup E] {f : ι → E},
(CauchySeq fun s => ∑ i ∈ s, f i) ↔ ∀ ε > 0, ∃ s, ∀ (t : Finset ι), Disjoint t s → ‖∑ i ∈ t, f i‖ < ε- Cited by
- 2 results in Mathlib
- Foundations
- Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SeminormedAddCommGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Finsetstatement and proof · cited by 13,712
- Norm.normstatement and proof · cited by 5,413
- Finset.sumstatement and proof · cited by 5,195
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Disjointstatement and proof · cited by 2,201
- CauchySeqstatement · cited by 131
- Metric.nhds_basis_ballproof · cited by 41
- Filter.HasBasis.forall_iffproof · cited by 15
- ball_zero_eqproof · cited by 8
- cauchySeq_finset_iff_sum_vanishingproof · cited by 5
Cited by2
Results whose statement or proof uses this declaration.
- summable_iff_vanishing_normproof · cited by 2
- cauchySeq_finset_of_norm_bounded_eventuallyproof · cited by 2