Theorems · Theorem · complex analysis
Complex.norm_le_norm_of_mapsTo_ball
∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [inst_2 : NormedAddCommGroup F]
[inst_3 : NormedSpace ℂ F] {R : ℝ} {f : E → F} {z : E},
DifferentiableOn ℂ f (Metric.ball 0 R) →
Set.MapsTo f (Metric.ball 0 R) (Metric.closedBall 0 R) → f 0 = 0 → ‖z‖ < R → ‖f z‖ ≤ ‖z‖The Schwarz Lemma.
Let f : E → F be a complex analytic on an open ball with center at the origin.
If f sends this ball to the closed ball with center 0 of the same radius and f 0 = 0,
then for any point z of this disk we have ‖f z‖ ≤ ‖z‖.
- Defined in
- Mathlib.Analysis.Complex.Schwarz
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 295 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- Complexstatement and proof · cited by 5,565
- Norm.normstatement and proof · cited by 5,413
- Dist.distproof · cited by 1,539
- Metric.ballstatement and proof · cited by 735
- Set.MapsTostatement and proof · cited by 732
- Metric.closedBallstatement and proof · cited by 704
- DifferentiableOnstatement and proof · cited by 419
- dist_zero_rightproof · cited by 172
- mem_ball_zero_iffproof · cited by 22
Cited by1
Results whose statement or proof uses this declaration.
- Complex.norm_le_norm_of_mapsTo_ball_selfproof · cited by 0