Theorems · Theorem · complex analysis

Differentiable.exists_const_forall_eq_of_bounded

∀ {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) → ∃ c, ∀ (z : E), f z = c

Liouville's theorem: a complex differentiable bounded function is a constant.

Defined in: Mathlib.Analysis.Complex.Liouville
Cited by: 1 results in Mathlib
Foundations: Depth 290 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup NormedSpace NormedAddCommGroup NormedSpace

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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
Differentiablestatement and proof · cited by 298
Bornology.IsBoundedstatement and proof · cited by 293
Differentiable.apply_eq_apply_of_boundedproof · cited by 1

Cited by1

Results whose statement or proof uses this declaration.

Differentiable.exists_eq_const_of_boundedproof · cited by 1