Theorems · Theorem · functional analysis
hasFDerivAt_exp_of_mem_ball
∀ {𝕂 : Type u_1} {𝔸 : Type u_2} [inst : NontriviallyNormedField 𝕂] [inst_1 : NormedCommRing 𝔸]
[inst_2 : NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] [CharZero 𝕂] {x : 𝔸},
x ∈ Metric.eball 0 (NormedSpace.expSeries 𝕂 𝔸).radius → HasFDerivAt NormedSpace.exp (NormedSpace.exp x • 1) xThe exponential map in a commutative Banach algebra 𝔸 over a normed field 𝕂 of
characteristic zero has Fréchet derivative NormedSpace.exp x • 1 : 𝔸 →L[𝕂] 𝔸
at any point x in the disk of convergence.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites33
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement · cited by 53,352
- RingHom.idstatement · cited by 18,349
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- nhdsproof · cited by 5,554
- ContinuousLinearMapstatement · cited by 5,352
- Filter.Eventuallyproof · cited by 3,134
- CompleteSpacestatement and proof · cited by 2,532
- Nat.cast_oneproof · cited by 2,501
- zero_addproof · cited by 2,366
- Filter.EventuallyEqproof · cited by 1,912
- Filter.univ_mem'proof · cited by 1,672
Cited by1
Results whose statement or proof uses this declaration.
- hasStrictFDerivAt_exp_of_mem_ballproof · cited by 2