Theorems · Theorem · complex analysis
Differentiable.apply_eq_apply_of_bounded
- 1000+ list: Liouville's theorem (complex analysis)
∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {F : Type v} [inst_2 : NormedAddCommGroup F]
[inst_3 : NormedSpace ℂ F] {f : E → F},
Differentiable ℂ f → Bornology.IsBounded (Set.range f) → ∀ (z w : E), f z = f wLiouville's theorem: a complex differentiable bounded function f : E → F is a constant.
- Defined in
- Mathlib.Analysis.Complex.Liouville
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 289 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- Complexstatement and proof · cited by 5,565
- Set.rangestatement and proof · cited by 4,705
- zero_addproof · cited by 2,366
- one_smulproof · cited by 1,374
- zero_smulproof · cited by 716
- sub_add_cancelproof · cited by 344
- Differentiablestatement and proof · cited by 298
- Bornology.IsBoundedstatement and proof · cited by 293
- Bornology.IsBounded.subsetproof · cited by 45
- differentiable_idproof · cited by 35
Cited by1
Results whose statement or proof uses this declaration.
- Differentiable.exists_const_forall_eq_of_boundedproof · cited by 1