Theorems · Theorem · complex analysis
FormalMultilinearSeries.radius_eq_liminf
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
(p : FormalMultilinearSeries 𝕜 E F), p.radius = Filter.liminf (fun n => 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) Filter.atTopThe radius of a formal multilinear series is equal to
$\liminf_{n\to\infty} \frac{1}{\sqrt[n]{‖p n‖}}$. The actual statement uses ℝ≥0 and some
coercions.
- Defined in
- Mathlib.Analysis.Analytic.RadiusLiminf
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites61
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
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- ENNRealstatement and proof · cited by 9,879
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Norm.normproof · cited by 5,413
- NNRealstatement and proof · cited by 4,310
- LE.le.transproof · cited by 3,151
- Filter.Eventuallyproof · cited by 3,134
- Filter.atTopstatement and proof · cited by 2,405
- mul_commproof · cited by 2,262
- LT.lt.leproof · cited by 2,189
Cited by2
Results whose statement or proof uses this declaration.
- spectrum.limsup_pow_nnnorm_pow_one_div_le_spectralRadiusproof · cited by 1
- FormalMultilinearSeries.radius_inv_eq_limsupproof · cited by 0