English
Another formulation: spectrum 𝕜 f equals the preimage under the algebra map of the range of f, for f: X → R with appropriate structure.
Русский
Ещё одно формулирование: спектр 𝕜 f равен предобразу под алгебраическим отображением диапазона f.
LaTeX
$$$\mathrm{spectrum}_{\mathbb{K}}(f) = (\mathrm{algebraMap}_{\mathbb{K}, R})^{-1}(\mathrm{range}(f)).$$$
Lean4
theorem spectrum_eq_preimage_range (f : C(X, R)) : spectrum 𝕜 f = algebraMap _ _ ⁻¹' Set.range f :=
by
ext x
simp only [spectrum.mem_iff, isUnit_iff_forall_ne_zero, not_forall, sub_apply, Classical.not_not, Set.mem_range,
sub_eq_zero, @eq_comm _ (x • 1 : R) _, Set.mem_preimage, Algebra.algebraMap_eq_smul_one, smul_apply, one_apply]