Theorems · Theorem · functional analysis

spectrum.map_polynomial_aeval

∀ {A : Type u_2} [inst : NormedRing A] [inst_1 : NormedAlgebra ℂ A] [CompleteSpace A] [Nontrivial A] (a : A)
  (p : Polynomial ℂ), spectrum ℂ ((Polynomial.aeval a) p) = (fun k => Polynomial.eval k p) '' spectrum ℂ a

The spectral mapping theorem for polynomials in a Banach algebra over ℂ.

Defined in

Mathlib.Analysis.Normed.Algebra.GelfandFormula

Cited by

1 results in Mathlib

Foundations

Depth 296 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedRingNormedAlgebraCompleteSpaceNontrivial