Theorems · Theorem · functional analysis

spectrum.hasFDerivAt_resolvent

∀ {𝕜 : Type u_1} {A : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedRing A] [inst_2 : NormedAlgebra 𝕜 A]
  [CompleteSpace A] {a : A} {k : 𝕜},
  k ∈ resolventSet 𝕜 a →
    HasFDerivAt (fun x => resolvent x k) (((ContinuousLinearMap.mulLeftRight 𝕜 A) (resolvent a k)) (resolvent a k)) a

Defined in

Mathlib.Analysis.Normed.Algebra.GelfandFormula

Cited by

1 results in Mathlib

Foundations

Depth 180 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedRingNormedAlgebraCompleteSpace