English
If 𝕜 is a field with 𝕜-algebra γ, then the C-map constant homomorphism equips α →ᵇ γ with a 𝕜-algebra structure extending the multi-valued algebra operations pointwise.
Русский
Если 𝕜 — поле с γ-алгеброй, то константная карта C делает пространство ограниченных функций α →ᵇ γ алгебраическим над 𝕜 в точечном произведении.
LaTeX
$$$\\text{C}: 𝕜 \\to^+* (α \\to^b γ)$ как RingHom with compatibility to scalar action.$$
Lean4
/-- `BoundedContinuousFunction.const` as a `RingHom`. -/
def C : 𝕜 →+* α →ᵇ γ where
toFun := fun c : 𝕜 => const α ((algebraMap 𝕜 γ) c)
map_one' := ext fun _ => (algebraMap 𝕜 γ).map_one
map_mul' _ _ := ext fun _ => (algebraMap 𝕜 γ).map_mul _ _
map_zero' := ext fun _ => (algebraMap 𝕜 γ).map_zero
map_add' _ _ := ext fun _ => (algebraMap 𝕜 γ).map_add _ _
