# Documentation

Mathlib.Topology.Algebra.ContinuousMonoidHom

# Continuous Monoid Homs #

This file defines the space of continuous homomorphisms between two topological groups.

## Main definitions #

• ContinuousMonoidHom A B: The continuous homomorphisms A →* B.
• ContinuousAddMonoidHom A B: The continuous additive homomorphisms A →+ B.
structure ContinuousAddMonoidHom (A : Type u_7) (B : Type u_8) [] [] [] [] extends :
Type (max u_7 u_8)

The type of continuous additive monoid homomorphisms from A to B.

Instances For
structure ContinuousMonoidHom (A : Type u_2) (B : Type u_3) [] [] [] [] extends :
Type (max u_2 u_3)

The type of continuous monoid homomorphisms from A to B.

When possible, instead of parametrizing results over (f : ContinuousMonoidHom A B), you should parametrize over (F : Type*) [ContinuousMonoidHomClass F A B] (f : F).

When you extend this structure, make sure to extend ContinuousAddMonoidHomClass.

Instances For
class ContinuousAddMonoidHomClass (F : Type u_1) (A : outParam (Type u_7)) (B : outParam (Type u_8)) [] [] [] [] extends :
Type (max (max u_1 u_7) u_8)
• coe : FAB
• coe_injective' : Function.Injective FunLike.coe
• map_add : ∀ (f : F) (x y : A), f (x + y) = f x + f y
• map_zero : ∀ (f : F), f 0 = 0
• map_continuous : ∀ (f : F),

Proof of the continuity of the map.

ContinuousAddMonoidHomClass F A B states that F is a type of continuous additive monoid homomorphisms.

You should also extend this typeclass when you extend ContinuousAddMonoidHom.

Instances
class ContinuousMonoidHomClass (F : Type u_1) (A : outParam (Type u_7)) (B : outParam (Type u_8)) [] [] [] [] extends :
Type (max (max u_1 u_7) u_8)
• coe : FAB
• coe_injective' : Function.Injective FunLike.coe
• map_mul : ∀ (f : F) (x y : A), f (x * y) = f x * f y
• map_one : ∀ (f : F), f 1 = 1
• map_continuous : ∀ (f : F),

Proof of the continuity of the map.

ContinuousMonoidHomClass F A B states that F is a type of continuous additive monoid homomorphisms.

You should also extend this typeclass when you extend ContinuousMonoidHom.

Instances
instance ContinuousAddMonoidHomClass.toContinuousMapClass (F : Type u_1) (A : Type u_2) (B : Type u_3) [] [] [] [] [] :
instance ContinuousMonoidHomClass.toContinuousMapClass (F : Type u_1) (A : Type u_2) (B : Type u_3) [] [] [] [] [] :
theorem ContinuousAddMonoidHom.ContinuousMonoidHom.ContinuousAddMonoidHomClass.proof_3 {A : Type u_2} {B : Type u_1} [] [] [] [] (f : ) :
ZeroHom.toFun (f.toAddMonoidHom) 0 = 0
theorem ContinuousAddMonoidHom.ContinuousMonoidHom.ContinuousAddMonoidHomClass.proof_1 {A : Type u_2} {B : Type u_1} [] [] [] [] (f : ) (g : ) (h : (fun f => f.toFun) f = (fun f => f.toFun) g) :
f = g
theorem ContinuousAddMonoidHom.ContinuousMonoidHom.ContinuousAddMonoidHomClass.proof_2 {A : Type u_2} {B : Type u_1} [] [] [] [] (f : ) (x : A) (y : A) :
ZeroHom.toFun (f.toAddMonoidHom) (x + y) = ZeroHom.toFun (f.toAddMonoidHom) x + ZeroHom.toFun (f.toAddMonoidHom) y
instance ContinuousAddMonoidHom.instCoeFunContinuousAddMonoidHomForAll {A : Type u_2} {B : Type u_3} [] [] [] [] :
CoeFun () fun x => AB

Helper instance for when there's too many metavariables to apply FunLike.hasCoeToFun.

instance ContinuousMonoidHom.instCoeFunContinuousMonoidHomForAll {A : Type u_2} {B : Type u_3} [] [] [] [] :
CoeFun () fun x => AB

Helper instance for when there's too many metavariables to apply FunLike.hasCoeToFun directly.

theorem ContinuousAddMonoidHom.ext {A : Type u_2} {B : Type u_3} [] [] [] [] {f : } {g : } (h : ∀ (x : A), f x = g x) :
f = g
theorem ContinuousMonoidHom.ext {A : Type u_2} {B : Type u_3} [] [] [] [] {f : } {g : } (h : ∀ (x : A), f x = g x) :
f = g
def ContinuousAddMonoidHom.toContinuousMap {A : Type u_2} {B : Type u_3} [] [] [] [] (f : ) :
C(A, B)

Reinterpret a ContinuousAddMonoidHom as a ContinuousMap.

Instances For
def ContinuousMonoidHom.toContinuousMap {A : Type u_2} {B : Type u_3} [] [] [] [] (f : ) :
C(A, B)

Reinterpret a ContinuousMonoidHom as a ContinuousMap.

Instances For
theorem ContinuousAddMonoidHom.toContinuousMap_injective {A : Type u_2} {B : Type u_3} [] [] [] [] :
theorem ContinuousMonoidHom.toContinuousMap_injective {A : Type u_2} {B : Type u_3} [] [] [] [] :
Function.Injective ContinuousMonoidHom.toContinuousMap
def ContinuousAddMonoidHom.mk' {A : Type u_2} {B : Type u_3} [] [] [] [] (f : A →+ B) (hf : ) :

Construct a ContinuousAddMonoidHom from a Continuous AddMonoidHom.

Instances For
theorem ContinuousAddMonoidHom.mk'.proof_1 {A : Type u_2} {B : Type u_1} [] [] (f : A →+ B) (x : A) (y : A) :
ZeroHom.toFun (f) (x + y) = ZeroHom.toFun (f) x + ZeroHom.toFun (f) y
def ContinuousMonoidHom.mk' {A : Type u_2} {B : Type u_3} [] [] [] [] (f : A →* B) (hf : ) :

Construct a ContinuousMonoidHom from a Continuous MonoidHom.

Instances For
def ContinuousAddMonoidHom.comp {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (g : ) (f : ) :

Composition of two continuous homomorphisms.

Instances For
theorem ContinuousAddMonoidHom.comp.proof_1 {A : Type u_2} {B : Type u_3} {C : Type u_1} [] [] [] [] [] [] (g : ) (f : ) :
Continuous (g.toFun fun x => f.toAddMonoidHom x)
@[simp]
theorem ContinuousAddMonoidHom.comp_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (g : ) (f : ) :
∀ (a : A), ↑() a = g.toAddMonoidHom (f.toAddMonoidHom a)
@[simp]
theorem ContinuousMonoidHom.comp_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (g : ) (f : ) :
∀ (a : A), ↑() a = g.toMonoidHom (f.toMonoidHom a)
def ContinuousMonoidHom.comp {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (g : ) (f : ) :

Composition of two continuous homomorphisms.

Instances For
def ContinuousAddMonoidHom.sum {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (f : ) (g : ) :

Product of two continuous homomorphisms on the same space.

Instances For
theorem ContinuousAddMonoidHom.sum.proof_1 {A : Type u_3} {B : Type u_2} {C : Type u_1} [] [] [] [] [] [] (f : ) (g : ) :
Continuous fun x => (ZeroHom.toFun (f.toAddMonoidHom) x, g.toAddMonoidHom x)
@[simp]
theorem ContinuousMonoidHom.prod_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (f : ) (g : ) (i : A) :
↑() i = (f.toMonoidHom i, g.toMonoidHom i)
@[simp]
theorem ContinuousAddMonoidHom.sum_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (f : ) (g : ) (i : A) :
↑() i = (f.toAddMonoidHom i, g.toAddMonoidHom i)
def ContinuousMonoidHom.prod {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (f : ) (g : ) :

Product of two continuous homomorphisms on the same space.

Instances For
def ContinuousAddMonoidHom.sum_map {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [] [] [] [] [] [] [] [] (f : ) (g : ) :

Product of two continuous homomorphisms on different spaces.

Instances For
theorem ContinuousAddMonoidHom.sum_map.proof_1 {A : Type u_3} {B : Type u_4} {C : Type u_2} {D : Type u_1} [] [] [] [] [] [] [] [] (f : ) (g : ) :
Continuous fun x => (ZeroHom.toFun (f.toAddMonoidHom) x.fst, (g.toAddMonoidHom).1 x.snd)
@[simp]
theorem ContinuousMonoidHom.prod_map_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [] [] [] [] [] [] [] [] (f : ) (g : ) (i : A × B) :
↑() i = (f.toMonoidHom i.fst, g.toMonoidHom i.snd)
@[simp]
theorem ContinuousAddMonoidHom.sum_map_toFun {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [] [] [] [] [] [] [] [] (f : ) (g : ) (i : A × B) :
↑() i = (f.toAddMonoidHom i.fst, g.toAddMonoidHom i.snd)
def ContinuousMonoidHom.prod_map {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [] [] [] [] [] [] [] [] (f : ) (g : ) :

Product of two continuous homomorphisms on different spaces.

Instances For
def ContinuousAddMonoidHom.zero (A : Type u_2) (B : Type u_3) [] [] [] [] :

The trivial continuous homomorphism.

Instances For
theorem ContinuousAddMonoidHom.zero.proof_1 (A : Type u_2) (B : Type u_1) [] [] [] :
Continuous fun x => AddZeroClass.toZero.1
@[simp]
theorem ContinuousAddMonoidHom.zero_toFun (A : Type u_2) (B : Type u_3) [] [] [] [] :
∀ (x : A), ↑() x = 0
@[simp]
theorem ContinuousMonoidHom.one_toFun (A : Type u_2) (B : Type u_3) [] [] [] [] :
∀ (x : A), ↑() x = 1
def ContinuousMonoidHom.one (A : Type u_2) (B : Type u_3) [] [] [] [] :

The trivial continuous homomorphism.

Instances For
instance ContinuousMonoidHom.instInhabitedContinuousMonoidHom (A : Type u_2) (B : Type u_3) [] [] [] [] :
def ContinuousAddMonoidHom.id (A : Type u_2) [] [] :

The identity continuous homomorphism.

Instances For
@[simp]
theorem ContinuousMonoidHom.id_toFun (A : Type u_2) [] [] (x : A) :
↑() x = x
@[simp]
theorem ContinuousAddMonoidHom.id_toFun (A : Type u_2) [] [] (x : A) :
= x
def ContinuousMonoidHom.id (A : Type u_2) [] [] :

The identity continuous homomorphism.

Instances For
def ContinuousAddMonoidHom.fst (A : Type u_2) (B : Type u_3) [] [] [] [] :

The continuous homomorphism given by projection onto the first factor.

Instances For
@[simp]
theorem ContinuousMonoidHom.fst_toFun (A : Type u_2) (B : Type u_3) [] [] [] [] (self : A × B) :
↑() self = self.fst
@[simp]
theorem ContinuousAddMonoidHom.fst_toFun (A : Type u_2) (B : Type u_3) [] [] [] [] (self : A × B) :
↑() self = self.fst
def ContinuousMonoidHom.fst (A : Type u_2) (B : Type u_3) [] [] [] [] :

The continuous homomorphism given by projection onto the first factor.

Instances For
def ContinuousAddMonoidHom.snd (A : Type u_2) (B : Type u_3) [] [] [] [] :

The continuous homomorphism given by projection onto the second factor.

Instances For
@[simp]
theorem ContinuousAddMonoidHom.snd_toFun (A : Type u_2) (B : Type u_3) [] [] [] [] (self : A × B) :
↑() self = self.snd
@[simp]
theorem ContinuousMonoidHom.snd_toFun (A : Type u_2) (B : Type u_3) [] [] [] [] (self : A × B) :
↑() self = self.snd
def ContinuousMonoidHom.snd (A : Type u_2) (B : Type u_3) [] [] [] [] :

The continuous homomorphism given by projection onto the second factor.

Instances For
def ContinuousAddMonoidHom.inl (A : Type u_2) (B : Type u_3) [] [] [] [] :

The continuous homomorphism given by inclusion of the first factor.

Instances For
@[simp]
theorem ContinuousMonoidHom.inl_toFun (A : Type u_2) (B : Type u_3) [] [] [] [] (i : A) :
↑() i = (().toMonoidHom i, ().toMonoidHom i)
@[simp]
theorem ContinuousAddMonoidHom.inl_toFun (A : Type u_2) (B : Type u_3) [] [] [] [] (i : A) :
↑() i = (().toAddMonoidHom i, ().toAddMonoidHom i)
def ContinuousMonoidHom.inl (A : Type u_2) (B : Type u_3) [] [] [] [] :

The continuous homomorphism given by inclusion of the first factor.

Instances For
def ContinuousAddMonoidHom.inr (A : Type u_2) (B : Type u_3) [] [] [] [] :

The continuous homomorphism given by inclusion of the second factor.

Instances For
@[simp]
theorem ContinuousMonoidHom.inr_toFun (A : Type u_2) (B : Type u_3) [] [] [] [] (i : B) :
↑() i = (().toMonoidHom i, ().toMonoidHom i)
@[simp]
theorem ContinuousAddMonoidHom.inr_toFun (A : Type u_2) (B : Type u_3) [] [] [] [] (i : B) :
↑() i = (().toAddMonoidHom i, ().toAddMonoidHom i)
def ContinuousMonoidHom.inr (A : Type u_2) (B : Type u_3) [] [] [] [] :

The continuous homomorphism given by inclusion of the second factor.

Instances For

The continuous homomorphism given by the diagonal embedding.

Instances For
@[simp]
theorem ContinuousMonoidHom.diag_toFun (A : Type u_2) [] [] (i : A) :
↑() i = (().toMonoidHom i, ().toMonoidHom i)
@[simp]
theorem ContinuousAddMonoidHom.diag_toFun (A : Type u_2) [] [] (i : A) :

The continuous homomorphism given by the diagonal embedding.

Instances For
def ContinuousAddMonoidHom.swap (A : Type u_2) (B : Type u_3) [] [] [] [] :

The continuous homomorphism given by swapping components.

Instances For
@[simp]
theorem ContinuousMonoidHom.swap_toFun (A : Type u_2) (B : Type u_3) [] [] [] [] (i : A × B) :
↑() i = (().toMonoidHom i, ().toMonoidHom i)
@[simp]
theorem ContinuousAddMonoidHom.swap_toFun (A : Type u_2) (B : Type u_3) [] [] [] [] (i : A × B) :
↑() i = (().toAddMonoidHom i, ().toAddMonoidHom i)
def ContinuousMonoidHom.swap (A : Type u_2) (B : Type u_3) [] [] [] [] :

The continuous homomorphism given by swapping components.

Instances For

The continuous homomorphism given by addition.

Instances For
theorem ContinuousAddMonoidHom.add.proof_1 (E : Type u_1) [] [] :
Continuous fun p => p.fst + p.snd
@[simp]
theorem ContinuousAddMonoidHom.add_toFun (E : Type u_6) [] [] :
∀ (a : E × E), = a.fst + a.snd
@[simp]
theorem ContinuousMonoidHom.mul_toFun (E : Type u_6) [] [] [] :
∀ (a : E × E), ↑() a = a.fst * a.snd
def ContinuousMonoidHom.mul (E : Type u_6) [] [] [] :

The continuous homomorphism given by multiplication.

Instances For
theorem ContinuousAddMonoidHom.neg.proof_1 (E : Type u_1) [] [] :
Continuous fun a => -a
def ContinuousAddMonoidHom.neg (E : Type u_6) [] [] :

The continuous homomorphism given by negation.

Instances For
@[simp]
theorem ContinuousAddMonoidHom.neg_toFun (E : Type u_6) [] [] :
∀ (a : E), = -a
@[simp]
theorem ContinuousMonoidHom.inv_toFun (E : Type u_6) [] [] [] :
∀ (a : E), ↑() a = a⁻¹
def ContinuousMonoidHom.inv (E : Type u_6) [] [] [] :

The continuous homomorphism given by inversion.

Instances For
def ContinuousAddMonoidHom.coprod {A : Type u_2} {B : Type u_3} {E : Type u_6} [] [] [] [] [] [] (f : ) (g : ) :

Coproduct of two continuous homomorphisms to the same space.

Instances For
@[simp]
theorem ContinuousAddMonoidHom.coprod_toFun {A : Type u_2} {B : Type u_3} {E : Type u_6} [] [] [] [] [] [] (f : ) (g : ) :
∀ (a : A × B), ↑() a = ().toAddMonoidHom (().toAddMonoidHom a)
@[simp]
theorem ContinuousMonoidHom.coprod_toFun {A : Type u_2} {B : Type u_3} {E : Type u_6} [] [] [] [] [] [] [] (f : ) (g : ) :
∀ (a : A × B), ↑() a = ().toMonoidHom (().toMonoidHom a)
def ContinuousMonoidHom.coprod {A : Type u_2} {B : Type u_3} {E : Type u_6} [] [] [] [] [] [] [] (f : ) (g : ) :

Coproduct of two continuous homomorphisms to the same space.

Instances For
∀ (a : ), zsmulRec 0 a = zsmulRec 0 a
∀ (n : ) (a : ), zsmulRec () a = zsmulRec () a
∀ (n : ) (a : ), zsmulRec () a = zsmulRec () a
∀ (n : ) (x : ), nsmulRec (n + 1) x = nsmulRec (n + 1) x
∀ (x : ), nsmulRec 0 x = nsmulRec 0 x
theorem ContinuousAddMonoidHom.inducing_toContinuousMap (A : Type u_2) (B : Type u_3) [] [] [] [] :
theorem ContinuousMonoidHom.inducing_toContinuousMap (A : Type u_2) (B : Type u_3) [] [] [] [] :
Inducing ContinuousMonoidHom.toContinuousMap
theorem ContinuousAddMonoidHom.embedding_toContinuousMap (A : Type u_2) (B : Type u_3) [] [] [] [] :
theorem ContinuousMonoidHom.embedding_toContinuousMap (A : Type u_2) (B : Type u_3) [] [] [] [] :
Embedding ContinuousMonoidHom.toContinuousMap
theorem ContinuousAddMonoidHom.closedEmbedding_toContinuousMap (A : Type u_2) (B : Type u_3) [] [] [] [] [] [] :
abbrev ContinuousAddMonoidHom.closedEmbedding_toContinuousMap.match_1 (A : Type u_1) (B : Type u_2) [] [] [] [] (f : ) (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (motive : {f | f x U} {f | f y V} {f | f (x + y) W}Prop) :
(x : {f | f x U} {f | f y V} {f | f (x + y) W}) → ((hfU : {f | f x U}) → (hfV : {f | f y V}) → (hfW : {f | f (x + y) W}) → motive (_ : {f | f x U} {f | f y V} {f | f (x + y) W})) → motive x
Instances For
theorem ContinuousMonoidHom.closedEmbedding_toContinuousMap (A : Type u_2) (B : Type u_3) [] [] [] [] [] [] :
ClosedEmbedding ContinuousMonoidHom.toContinuousMap
theorem ContinuousAddMonoidHom.continuous_of_continuous_uncurry {B : Type u_3} {C : Type u_4} [] [] [] [] {A : Type u_7} [] (f : A) (h : Continuous (Function.uncurry fun x y => ↑(f x) y)) :
theorem ContinuousMonoidHom.continuous_of_continuous_uncurry {B : Type u_3} {C : Type u_4} [] [] [] [] {A : Type u_7} [] (f : A) (h : Continuous (Function.uncurry fun x y => ↑(f x) y)) :
theorem ContinuousAddMonoidHom.continuous_comp {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] :
theorem ContinuousMonoidHom.continuous_comp {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] :
theorem ContinuousAddMonoidHom.continuous_comp_left {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (f : ) :
Continuous fun g =>
theorem ContinuousMonoidHom.continuous_comp_left {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (f : ) :
Continuous fun g =>
theorem ContinuousAddMonoidHom.continuous_comp_right {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (f : ) :
Continuous fun g =>
theorem ContinuousMonoidHom.continuous_comp_right {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (f : ) :
Continuous fun g =>
theorem ContinuousAddMonoidHom.compLeft.proof_2 {A : Type u_3} {B : Type u_2} (E : Type u_1) [] [] [] [] [] [] (f : ) (_g : ) (_h : ) :
ZeroHom.toFun { toFun := fun g => , map_zero' := (_ : (fun g => ) 0 = (fun g => ) 0) } (_g + _h) = ZeroHom.toFun { toFun := fun g => , map_zero' := (_ : (fun g => ) 0 = (fun g => ) 0) } (_g + _h)
theorem ContinuousAddMonoidHom.compLeft.proof_3 {A : Type u_2} {B : Type u_3} (E : Type u_1) [] [] [] [] [] [] (f : ) :
Continuous fun g =>
theorem ContinuousAddMonoidHom.compLeft.proof_1 {A : Type u_1} {B : Type u_3} (E : Type u_2) [] [] [] [] [] [] (f : ) :
(fun g => ) 0 = (fun g => ) 0
def ContinuousAddMonoidHom.compLeft {A : Type u_2} {B : Type u_3} (E : Type u_6) [] [] [] [] [] [] (f : ) :

ContinuousAddMonoidHom _ f is a functor.

Instances For
def ContinuousMonoidHom.compLeft {A : Type u_2} {B : Type u_3} (E : Type u_6) [] [] [] [] [] [] [] (f : ) :

ContinuousMonoidHom _ f is a functor.

Instances For
theorem ContinuousAddMonoidHom.compRight.proof_2 (A : Type u_2) {E : Type u_3} [] [] [] [] {B : Type u_1} [] [] (f : ) (g : ) (h : ) :
ZeroHom.toFun { toFun := fun g => , map_zero' := (_ : (fun g => ) 0 = 0) } (g + h) = ZeroHom.toFun { toFun := fun g => , map_zero' := (_ : (fun g => ) 0 = 0) } g + ZeroHom.toFun { toFun := fun g => , map_zero' := (_ : (fun g => ) 0 = 0) } h
theorem ContinuousAddMonoidHom.compRight.proof_3 (A : Type u_2) {E : Type u_1} [] [] [] [] {B : Type u_3} [] [] (f : ) :
Continuous fun g =>
theorem ContinuousAddMonoidHom.compRight.proof_1 (A : Type u_1) {E : Type u_2} [] [] [] [] {B : Type u_3} [] [] (f : ) :
(fun g => ) 0 = 0
def ContinuousAddMonoidHom.compRight (A : Type u_2) {E : Type u_6} [] [] [] [] {B : Type u_7} [] [] (f : ) :

ContinuousAddMonoidHom f _ is a functor.

Instances For
def ContinuousMonoidHom.compRight (A : Type u_2) {E : Type u_6} [] [] [] [] [] {B : Type u_7} [] [] [] (f : ) :

ContinuousMonoidHom f _ is a functor.

Instances For
def PontryaginDual (A : Type u_2) [] [] :
Type u_2

The Pontryagin dual of A is the group of continuous homomorphism A → circle.

Instances For
noncomputable instance instCommGroupPontryaginDual (A : Type u_2) [] [] :
noncomputable instance instInhabitedPontryaginDual (A : Type u_2) [] [] :
noncomputable def PontryaginDual.map {A : Type u_2} {B : Type u_3} [] [] [] [] (f : ) :

PontryaginDual is a functor.

Instances For
@[simp]
theorem PontryaginDual.map_apply {A : Type u_2} {B : Type u_3} [] [] [] [] (f : ) (x : ) (y : A) :
↑(↑() x) y = x (f y)
@[simp]
theorem PontryaginDual.map_one {A : Type u_2} {B : Type u_3} [] [] [] [] :
@[simp]
theorem PontryaginDual.map_comp {A : Type u_2} {B : Type u_3} {C : Type u_4} [] [] [] [] [] [] (g : ) (f : ) :
@[simp]
theorem PontryaginDual.map_mul {A : Type u_2} {E : Type u_6} [] [] [] [] [] (f : ) (g : ) :
noncomputable def PontryaginDual.mapHom (A : Type u_2) (E : Type u_6) [] [] [] [] [] :

ContinuousMonoidHom.dual as a ContinuousMonoidHom.

Instances For