Theorems · Theorem · functional analysis
lp.tendsto_lp_of_tendsto_pi
∀ {α : Type u_3} {E : α → Type u_4} {p : ENNReal} [inst : (i : α) → NormedAddCommGroup (E i)] [_i : Fact (1 ≤ p)]
{F : ℕ → ↥(lp E p)},
CauchySeq F →
∀ {f : ↥(lp E p)},
Filter.Tendsto (id fun i => ↑(F i)) Filter.atTop (nhds ↑f) → Filter.Tendsto F Filter.atTop (nhds f)If a sequence is Cauchy in the lp E p topology and pointwise convergent to an element f of
lp E p, then it converges to f in the lp E p topology.
- Defined in
- Mathlib.Analysis.Normed.Lp.lpSpace
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 226 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NormedAddCommGroupFact
Around this declaration
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Cites27
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
- ENNRealstatement and proof · cited by 9,879
- Set.ofPredproof · cited by 6,101
- nhdsstatement and proof · cited by 5,554
- Norm.normproof · cited by 5,413
- Filter.Tendstostatement and proof · cited by 3,814
- AddSubgroupstatement · cited by 3,232
- Filter.Eventuallyproof · cited by 3,134
- Factstatement and proof · cited by 2,726
- Filter.atTopstatement and proof · cited by 2,405
- LT.lt.leproof · cited by 2,189
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