Theorems · Theorem · functional analysis

spectrum.smul_eq_smul

∀ {𝕜 : Type u} {A : Type v} [inst : Field 𝕜] [inst_1 : Ring A] [inst_2 : Algebra 𝕜 A] [Nontrivial A] (k : 𝕜) (a : A),
  (spectrum 𝕜 a).Nonempty → spectrum 𝕜 (k • a) = k • spectrum 𝕜 a

the assumption (σ a).Nonempty is necessary and cannot be removed without further conditions on the algebra A and scalar field 𝕜.

Defined in

Mathlib.Algebra.Algebra.Spectrum.Basic

Cited by

0 results in Mathlib

Foundations

Depth 59 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldRingAlgebraNontrivial