File: Mathlib/Topology/Algebra/Module/CharacterSpace.lean

full_name: WeakDual.characterSpace
code: /-- The character space of a topological algebra is the subset of elements of the weak dual that
are also algebra homomorphisms. -/
def characterSpace (𝕜 : Type*) (A : Type*) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜]
    [ContinuousConstSMul 𝕜 𝕜] [NonUnitalNonAssocSemiring A] [TopologicalSpace A] [Module 𝕜 A] 
goal: Set (WeakDual 𝕜 A)

full_name: WeakDual.CharacterSpace.instFunLike
code: instance instFunLike : FunLike (characterSpace 𝕜 A) A 𝕜 
goal: FunLike (↑(WeakDual.characterSpace 𝕜 A)) A 𝕜

full_name: WeakDual.CharacterSpace.instContinuousLinearMapClass
code: /-- Elements of the character space are continuous linear maps. -/
instance instContinuousLinearMapClass : ContinuousLinearMapClass (characterSpace 𝕜 A) 𝕜 A 𝕜 
goal: ContinuousLinearMapClass (↑(WeakDual.characterSpace 𝕜 A)) 𝕜 A 𝕜

full_name: WeakDual.CharacterSpace.coe_coe
code: @[simp, norm_cast]
protected theorem coe_coe (φ : characterSpace 𝕜 A) : ⇑(φ : WeakDual 𝕜 A) = (φ : A → 𝕜) 
goal: ⇑↑φ = ⇑φ

full_name: WeakDual.CharacterSpace.ext
code: @[ext]
theorem ext {φ ψ : characterSpace 𝕜 A} (h : ∀ x, φ x = ψ x) : φ = ψ 
goal: φ = ψ

full_name: WeakDual.CharacterSpace.toCLM
code: /-- An element of the character space, as a continuous linear map. -/
def toCLM (φ : characterSpace 𝕜 A) : A →L[𝕜] 𝕜 
goal: A →L[𝕜] 𝕜

full_name: WeakDual.CharacterSpace.coe_toCLM
code: @[simp]
theorem coe_toCLM (φ : characterSpace 𝕜 A) : ⇑(toCLM φ) = φ 
goal: ⇑(WeakDual.CharacterSpace.toCLM φ) = ⇑φ

full_name: WeakDual.CharacterSpace.instNonUnitalAlgHomClass
code: /-- Elements of the character space are non-unital algebra homomorphisms. -/
instance instNonUnitalAlgHomClass : NonUnitalAlgHomClass (characterSpace 𝕜 A) 𝕜 A 𝕜 
goal: NonUnitalAlgHomClass (↑(WeakDual.characterSpace 𝕜 A)) 𝕜 A 𝕜

full_name: WeakDual.CharacterSpace.toNonUnitalAlgHom
code: /-- An element of the character space, as a non-unital algebra homomorphism. -/
def toNonUnitalAlgHom (φ : characterSpace 𝕜 A) : A →ₙₐ[𝕜] 𝕜 
goal: A →ₙₐ[𝕜] 𝕜

full_name: WeakDual.CharacterSpace.coe_toNonUnitalAlgHom
code: @[simp]
theorem coe_toNonUnitalAlgHom (φ : characterSpace 𝕜 A) : ⇑(toNonUnitalAlgHom φ) = φ 
goal: ⇑(WeakDual.CharacterSpace.toNonUnitalAlgHom φ) = ⇑φ

full_name: WeakDual.CharacterSpace.instIsEmpty
code: instance instIsEmpty [Subsingleton A] : IsEmpty (characterSpace 𝕜 A) 
goal: IsEmpty ↑(WeakDual.characterSpace 𝕜 A)

full_name: WeakDual.CharacterSpace.union_zero
code: theorem union_zero :
    characterSpace 𝕜 A ∪ {0} = {φ : WeakDual 𝕜 A | ∀ x y : A, φ (x * y) = φ x * φ y} 
goal: WeakDual.characterSpace 𝕜 A ∪ {0} = {φ | ∀ (x y : A), φ (x * y) = φ x * φ y}

full_name: WeakDual.CharacterSpace.union_zero_isClosed
code: /-- The `characterSpace 𝕜 A` along with `0` is always a closed set in `WeakDual 𝕜 A`. -/
theorem union_zero_isClosed [T2Space 𝕜] [ContinuousMul 𝕜] :
    IsClosed (characterSpace 𝕜 A ∪ {0}) 
goal: IsClosed (WeakDual.characterSpace 𝕜 A ∪ {0})

full_name: WeakDual.CharacterSpace.instAlgHomClass
code: /-- In a unital algebra, elements of the character space are algebra homomorphisms. -/
instance instAlgHomClass : AlgHomClass (characterSpace 𝕜 A) 𝕜 A 𝕜 
goal: AlgHomClass (↑(WeakDual.characterSpace 𝕜 A)) 𝕜 A 𝕜

full_name: WeakDual.CharacterSpace.toAlgHom
code: /-- An element of the character space of a unital algebra, as an algebra homomorphism. -/
@[simps]
def toAlgHom (φ : characterSpace 𝕜 A) : A →ₐ[𝕜] 𝕜 
goal: A →ₐ[𝕜] 𝕜

full_name: WeakDual.CharacterSpace.toAlgHom_apply
code: /-- An element of the character space of a unital algebra, as an algebra homomorphism. -/
@[simps]
def toAlgHom (φ : characterSpace 𝕜 A) : A →ₐ[𝕜] 𝕜 
goal: ∀ (a : A), (WeakDual.CharacterSpace.toAlgHom φ) a = (WeakDual.CharacterSpace.toNonUnitalAlgHom φ).toFun a

full_name: WeakDual.CharacterSpace.eq_set_map_one_map_mul
code: theorem eq_set_map_one_map_mul [Nontrivial 𝕜] :
    characterSpace 𝕜 A = {φ : WeakDual 𝕜 A | φ 1 = 1 ∧ ∀ x y : A, φ (x * y) = φ x * φ y} 
goal: WeakDual.characterSpace 𝕜 A = {φ | φ 1 = 1 ∧ ∀ (x y : A), φ (x * y) = φ x * φ y}

full_name: WeakDual.CharacterSpace.isClosed
code: /-- under suitable mild assumptions on `𝕜`, the character space is a closed set in
`WeakDual 𝕜 A`. -/
protected theorem isClosed [Nontrivial 𝕜] [T2Space 𝕜] [ContinuousMul 𝕜] :
    IsClosed (characterSpace 𝕜 A) 
goal: IsClosed (WeakDual.characterSpace 𝕜 A)

full_name: WeakDual.CharacterSpace.apply_mem_spectrum
code: theorem apply_mem_spectrum [Nontrivial 𝕜] (φ : characterSpace 𝕜 A) (a : A) : φ a ∈ spectrum 𝕜 a 
goal: φ a ∈ spectrum 𝕜 a

full_name: WeakDual.CharacterSpace.ext_ker
code: theorem ext_ker {φ ψ : characterSpace 𝕜 A} (h : RingHom.ker φ = RingHom.ker ψ) : φ = ψ 
goal: φ = ψ

full_name: WeakDual.ker_isMaximal
code: /-- The `RingHom.ker` of `φ : characterSpace 𝕜 A` is maximal. -/
instance ker_isMaximal (φ : characterSpace 𝕜 A) : (RingHom.ker φ).IsMaximal 
goal: (RingHom.ker φ).IsMaximal

full_name: WeakDual.gelfandTransform_apply_apply
code: /-- The **Gelfand transform** is an algebra homomorphism (over `𝕜`) from a topological `𝕜`-algebra
`A` into the `𝕜`-algebra of continuous `𝕜`-valued functions on the `characterSpace 𝕜 A`.
The character space itself consists of all algebra homomorphisms from `A` to `𝕜`.  -/
@[simps]
def gelfandTransform : A →ₐ[𝕜] C(characterSpace 𝕜 A, 𝕜) 
goal: ((WeakDual.gelfandTransform 𝕜 A) a) φ = φ a

full_name: WeakDual.gelfandTransform
code: /-- The **Gelfand transform** is an algebra homomorphism (over `𝕜`) from a topological `𝕜`-algebra
`A` into the `𝕜`-algebra of continuous `𝕜`-valued functions on the `characterSpace 𝕜 A`.
The character space itself consists of all algebra homomorphisms from `A` to `𝕜`.  -/
@[simps]
def gelfandTransform : A →ₐ[𝕜] C(characterSpace 𝕜 A, 𝕜) 
goal: A →ₐ[𝕜] C(↑(WeakDual.characterSpace 𝕜 A), 𝕜)