Theorems · Theorem · functional analysis
lp.hasSum_single
∀ {α : Type u_3} {E : α → Type u_4} {p : ENNReal} [inst : (i : α) → NormedAddCommGroup (E i)] [inst_1 : DecidableEq α]
[inst_2 : Fact (1 ≤ p)], p ≠ ⊤ → ∀ (f : ↥(lp E p)), HasSum (fun i => lp.single p i (↑f i)) fThe canonical finitely-supported approximations to an element f of lp converge to it, in the
lp topology.
- Defined in
- Mathlib.Analysis.Normed.Lp.lpSpace
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 224 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites41
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realproof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- Finsetproof · cited by 13,712
- ENNRealstatement and proof · cited by 9,879
- Top.topstatement and proof · cited by 9,680
- Norm.normproof · cited by 5,413
- Finset.sumproof · cited by 5,195
- AddSubgroupstatement · cited by 3,232
- Factstatement and proof · cited by 2,726
- Finset.sum_congrproof · cited by 2,323
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- absproof · cited by 1,814
Cited by2
Results whose statement or proof uses this declaration.
- lp.ext_continuousAddMonoidHomproof · cited by 2
- HilbertBasis.hasSum_repr_symmproof · cited by 1