# Documentation

Mathlib.Topology.Algebra.Algebra

# Topological (sub)algebras #

A topological algebra over a topological semiring R is a topological semiring with a compatible continuous scalar multiplication by elements of R. We reuse typeclass ContinuousSMul for topological algebras.

## Results #

This is just a minimal stub for now!

The topological closure of a subalgebra is still a subalgebra, which as an algebra is a topological algebra.

theorem continuous_algebraMap_iff_smul (R : Type u_1) (A : Type u) [] [] [Algebra R A] [] [] :
Continuous ↑() Continuous fun p => p.fst p.snd
theorem continuous_algebraMap (R : Type u_1) (A : Type u) [] [] [Algebra R A] [] [] [] :
theorem continuousSMul_of_algebraMap (R : Type u_1) (A : Type u) [] [] [Algebra R A] [] [] (h : Continuous ↑()) :
@[simp]
theorem algebraMapClm_apply (R : Type u_1) (A : Type u) [] [] [Algebra R A] [] [] [] (a : R) :
↑() a = ↑() a
@[simp]
theorem algebraMapClm_toFun (R : Type u_1) (A : Type u) [] [] [Algebra R A] [] [] [] (a : R) :
↑() a = ↑() a
def algebraMapClm (R : Type u_1) (A : Type u) [] [] [Algebra R A] [] [] [] :
R →L[R] A

The inclusion of the base ring in a topological algebra as a continuous linear map.

Instances For
theorem algebraMapClm_coe (R : Type u_1) (A : Type u) [] [] [Algebra R A] [] [] [] :
↑() = ↑()
theorem algebraMapClm_toLinearMap (R : Type u_1) (A : Type u) [] [] [Algebra R A] [] [] [] :
↑() =
instance Subalgebra.continuousSMul {R : Type u_1} [] {A : Type u} [] [] [Algebra R A] [] [] (s : ) :
ContinuousSMul R { x // x s }
def Subalgebra.topologicalClosure {R : Type u_1} [] {A : Type u} [] [] [Algebra R A] (s : ) :

The closure of a subalgebra in a topological algebra as a subalgebra.

Instances For
@[simp]
theorem Subalgebra.topologicalClosure_coe {R : Type u_1} [] {A : Type u} [] [] [Algebra R A] (s : ) :
instance Subalgebra.topologicalSemiring {R : Type u_1} [] {A : Type u} [] [] [Algebra R A] (s : ) :
theorem Subalgebra.le_topologicalClosure {R : Type u_1} [] {A : Type u} [] [] [Algebra R A] (s : ) :
theorem Subalgebra.isClosed_topologicalClosure {R : Type u_1} [] {A : Type u} [] [] [Algebra R A] (s : ) :
theorem Subalgebra.topologicalClosure_minimal {R : Type u_1} [] {A : Type u} [] [] [Algebra R A] (s : ) {t : } (h : s t) (ht : IsClosed t) :
def Subalgebra.commSemiringTopologicalClosure {R : Type u_1} [] {A : Type u} [] [] [Algebra R A] [] (s : ) (hs : ∀ (x y : { x // x s }), x * y = y * x) :
CommSemiring { x // }

If a subalgebra of a topological algebra is commutative, then so is its topological closure.

Instances For
theorem Subalgebra.topologicalClosure_comap_homeomorph {R : Type u_1} [] {A : Type u} [] [] [Algebra R A] (s : ) {B : Type u_2} [] [Ring B] [] [Algebra R B] (f : B →ₐ[R] A) (f' : B ≃ₜ A) (w : f = f') :

This is really a statement about topological algebra isomorphisms, but we don't have those, so we use the clunky approach of talking about an algebra homomorphism, and a separate homeomorphism, along with a witness that as functions they are the same.

@[reducible]
def Subalgebra.commRingTopologicalClosure {R : Type u_1} [] {A : Type u} [] [Ring A] [Algebra R A] [] [] (s : ) (hs : ∀ (x y : { x // x s }), x * y = y * x) :
CommRing { x // }

If a subalgebra of a topological algebra is commutative, then so is its topological closure. See note [reducible non-instances].

Instances For
def Algebra.elementalAlgebra (R : Type u_1) [] {A : Type u} [] [Ring A] [Algebra R A] [] (x : A) :

The topological closure of the subalgebra generated by a single element.

Instances For
theorem Algebra.self_mem_elementalAlgebra (R : Type u_1) [] {A : Type u} [] [Ring A] [Algebra R A] [] (x : A) :
instance DivisionRing.continuousConstSMul_rat {A : Type u_1} [] [] [] [] :

The action induced by algebraRat is continuous.