Documentation

Mathlib.Topology.ContinuousMap.ContinuousMapZero

Continuous maps sending zero to zero #

This is the type of continuous maps from X to R such that (0 : X) ↦ (0 : R) for which we provide the scoped notation C(X, R)₀. We provide this as a dedicated type solely for the non-unital continuous functional calculus, as using various terms of type Ideal C(X, R) were overly burdensome on type class synthesis.

Of course, one could generalize to maps between pointed topological spaces, but that goes beyond the purpose of this type.

structure ContinuousMapZero (X : Type u_1) (R : Type u_2) [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] extends C(X, R) :

Type (max u_1 u_2)

The type of continuous maps which map zero to zero.

Note that one should never use the structure projection ContinuousMapZero.toContinuousMap and instead favor the coercion (↑) : C(X, R)₀ → C(X, R) available from the instance of ContinuousMapClass. All the instances on C(X, R)₀ from C(X, R) passes through this coercion, not the structure projection. Of course, the two are definitionally equal, but not reducibly so.

toFun : X → R
continuous_toFun : Continuous self.toFun
map_zero' : self.toContinuousMap 0 = 0

Instances For

def ContinuousMapZero.«termC(_,_)₀» :

Lean.ParserDescr

The type of continuous maps which map zero to zero.

Note that one should never use the structure projection ContinuousMapZero.toContinuousMap and instead favor the coercion (↑) : C(X, R)₀ → C(X, R) available from the instance of ContinuousMapClass. All the instances on C(X, R)₀ from C(X, R) passes through this coercion, not the structure projection. Of course, the two are definitionally equal, but not reducibly so.

Equations

One or more equations did not get rendered due to their size.

Instances For

instance ContinuousMapZero.instFunLike {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] :

FunLike (ContinuousMapZero X R) X R

Equations

ContinuousMapZero.instFunLike = { coe := fun (f : ContinuousMapZero X R) => f.toFun, coe_injective' := ⋯ }

instance ContinuousMapZero.instContinuousMapClass {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] :

ContinuousMapClass (ContinuousMapZero X R) X R

Equations

⋯ = ⋯

instance ContinuousMapZero.instZeroHomClass {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] :

ZeroHomClass (ContinuousMapZero X R) X R

Equations

⋯ = ⋯

theorem ContinuousMapZero.ext {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] {f g : ContinuousMapZero X R} (h : ∀ (x : X), f x = g x) :

f = g

@[simp]

theorem ContinuousMapZero.coe_mk {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] {f : C(X, R)} {h0 : f 0 = 0} :

⇑{ toContinuousMap := f, map_zero' := h0 } = ⇑f

theorem ContinuousMapZero.toContinuousMap_injective {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] :

Function.Injective toContinuousMap

theorem ContinuousMapZero.range_toContinuousMap {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] :

Set.range toContinuousMap = {f : C(X, R) | f 0 = 0}

def ContinuousMapZero.comp {X : Type u_1} {Y : Type u_2} {R : Type u_3} [Zero X] [Zero Y] [Zero R] [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace R] (g : ContinuousMapZero Y R) (f : ContinuousMapZero X Y) :

ContinuousMapZero X R

Composition of continuous maps which map zero to zero.

Equations

g.comp f = { toContinuousMap := (↑g).comp ↑f, map_zero' := ⋯ }

Instances For

@[simp]

theorem ContinuousMapZero.comp_apply {X : Type u_1} {Y : Type u_2} {R : Type u_3} [Zero X] [Zero Y] [Zero R] [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace R] (g : ContinuousMapZero Y R) (f : ContinuousMapZero X Y) (x : X) :

(g.comp f) x = g (f x)

instance ContinuousMapZero.instPartialOrder {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [PartialOrder R] :

PartialOrder (ContinuousMapZero X R)

Equations

ContinuousMapZero.instPartialOrder = PartialOrder.lift DFunLike.coe ⋯

theorem ContinuousMapZero.le_def {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [PartialOrder R] (f g : ContinuousMapZero X R) :

f ≤ g ↔ ∀ (x : X), f x ≤ g x

instance ContinuousMapZero.instTopologicalSpace {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] :

TopologicalSpace (ContinuousMapZero X R)

Equations

ContinuousMapZero.instTopologicalSpace = TopologicalSpace.induced toContinuousMap inferInstance

theorem ContinuousMapZero.isEmbedding_toContinuousMap {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] :

Topology.IsEmbedding toContinuousMap

@[deprecated ContinuousMapZero.isEmbedding_toContinuousMap]

theorem ContinuousMapZero.embedding_toContinuousMap {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] :

Topology.IsEmbedding toContinuousMap

Alias of ContinuousMapZero.isEmbedding_toContinuousMap.

instance ContinuousMapZero.instT0Space {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [T0Space R] :

T0Space (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instR0Space {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [R0Space R] :

R0Space (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instT1Space {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [T1Space R] :

T1Space (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instR1Space {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [R1Space R] :

R1Space (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instT2Space {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [T2Space R] :

T2Space (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instRegularSpace {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [RegularSpace R] :

RegularSpace (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instT3Space {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [T3Space R] :

T3Space (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instContinuousEvalConst {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] :

ContinuousEvalConst (ContinuousMapZero X R) X R

Equations

⋯ = ⋯

instance ContinuousMapZero.instContinuousEval {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [LocallyCompactPair X R] :

ContinuousEval (ContinuousMapZero X R) X R

Equations

⋯ = ⋯

theorem ContinuousMapZero.isClosedEmbedding_toContinuousMap {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [T1Space R] :

Topology.IsClosedEmbedding toContinuousMap

@[deprecated ContinuousMapZero.isClosedEmbedding_toContinuousMap]

theorem ContinuousMapZero.closedEmbedding_toContinuousMap {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [T1Space R] :

Topology.IsClosedEmbedding toContinuousMap

Alias of ContinuousMapZero.isClosedEmbedding_toContinuousMap.

theorem ContinuousMapZero.continuous_comp_left {X : Type u_4} {Y : Type u_5} {Z : Type u_6} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [Zero X] [Zero Y] [Zero Z] (f : ContinuousMapZero X Y) :

Continuous fun (g : ContinuousMapZero Y Z) => g.comp f

def ContinuousMapZero.id {R : Type u_3} [Zero R] [TopologicalSpace R] {s : Set R} [Zero ↑s] (h0 : ↑0 = 0) :

ContinuousMapZero (↑s) R

The identity function as an element of C(s, R)₀ when 0 ∈ (s : Set R).

Equations

ContinuousMapZero.id h0 = { toContinuousMap := ContinuousMap.restrict s (ContinuousMap.id R), map_zero' := h0 }

Instances For

@[simp]

theorem ContinuousMapZero.id_toFun {R : Type u_3} [Zero R] [TopologicalSpace R] {s : Set R} [Zero ↑s] (h0 : ↑0 = 0) (a✝ : ↑s) :

(ContinuousMapZero.id h0) a✝ = ↑a✝

@[simp]

theorem ContinuousMapZero.toContinuousMap_id {R : Type u_3} [Zero R] [TopologicalSpace R] {s : Set R} [Zero ↑s] (h0 : ↑0 = 0) :

↑(ContinuousMapZero.id h0) = ContinuousMap.restrict s (ContinuousMap.id R)

instance ContinuousMapZero.instZero {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [Zero R] :

Zero (ContinuousMapZero X R)

Equations

ContinuousMapZero.instZero = { zero := { toContinuousMap := 0, map_zero' := ⋯ } }

@[simp]

theorem ContinuousMapZero.coe_zero {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [Zero R] :

⇑0 = 0

instance ContinuousMapZero.instAdd {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [AddZeroClass R] [ContinuousAdd R] :

Add (ContinuousMapZero X R)

Equations

ContinuousMapZero.instAdd = { add := fun (f g : ContinuousMapZero X R) => { toContinuousMap := ↑f + ↑g, map_zero' := ⋯ } }

@[simp]

theorem ContinuousMapZero.coe_add {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [AddZeroClass R] [ContinuousAdd R] (f g : ContinuousMapZero X R) :

⇑(f + g) = ⇑f + ⇑g

instance ContinuousMapZero.instMul {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [MulZeroClass R] [ContinuousMul R] :

Mul (ContinuousMapZero X R)

Equations

ContinuousMapZero.instMul = { mul := fun (f g : ContinuousMapZero X R) => { toContinuousMap := ↑f * ↑g, map_zero' := ⋯ } }

@[simp]

theorem ContinuousMapZero.coe_mul {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [MulZeroClass R] [ContinuousMul R] (f g : ContinuousMapZero X R) :

⇑(f * g) = ⇑f * ⇑g

instance ContinuousMapZero.instSMul {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] {M : Type u_3} [Zero R] [SMulZeroClass M R] [ContinuousConstSMul M R] :

SMul M (ContinuousMapZero X R)

Equations

ContinuousMapZero.instSMul = { smul := fun (m : M) (f : ContinuousMapZero X R) => { toContinuousMap := m • ↑f, map_zero' := ⋯ } }

@[simp]

theorem ContinuousMapZero.coe_smul {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] {M : Type u_3} [Zero R] [SMulZeroClass M R] [ContinuousConstSMul M R] (m : M) (f : ContinuousMapZero X R) :

⇑(m • f) = m • ⇑f

instance ContinuousMapZero.instNonUnitalCommSemiring {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] :

NonUnitalCommSemiring (ContinuousMapZero X R)

Equations

ContinuousMapZero.instNonUnitalCommSemiring = Function.Injective.nonUnitalCommSemiring toContinuousMap ⋯ ⋯ ⋯ ⋯ ⋯

instance ContinuousMapZero.instModule {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] {M : Type u_3} [Semiring M] [Module M R] [ContinuousConstSMul M R] :

Module M (ContinuousMapZero X R)

Equations

ContinuousMapZero.instModule = Function.Injective.module M { toFun := fun (f : ContinuousMapZero X R) => { toFun := ⇑f, continuous_toFun := ⋯ }, map_zero' := ⋯, map_add' := ⋯ } ⋯ ⋯

instance ContinuousMapZero.instSMulCommClass {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] {M : Type u_3} {N : Type u_4} [SMulZeroClass M R] [ContinuousConstSMul M R] [SMulZeroClass N R] [ContinuousConstSMul N R] [SMulCommClass M N R] :

SMulCommClass M N (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instSMulCommClass' {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] {M : Type u_3} [SMulZeroClass M R] [SMulCommClass M R R] [ContinuousConstSMul M R] :

SMulCommClass M (ContinuousMapZero X R) (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instIsScalarTower {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] {M : Type u_3} {N : Type u_4} [SMulZeroClass M R] [ContinuousConstSMul M R] [SMulZeroClass N R] [ContinuousConstSMul N R] [SMul M N] [IsScalarTower M N R] :

IsScalarTower M N (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instIsScalarTower' {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] {M : Type u_3} [SMulZeroClass M R] [IsScalarTower M R R] [ContinuousConstSMul M R] :

IsScalarTower M (ContinuousMapZero X R) (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instStarRing {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] [StarRing R] [ContinuousStar R] :

StarRing (ContinuousMapZero X R)

Equations

ContinuousMapZero.instStarRing = StarRing.mk ⋯

instance ContinuousMapZero.instStarModule {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] [StarRing R] {M : Type u_3} [SMulZeroClass M R] [ContinuousConstSMul M R] [Star M] [StarModule M R] [ContinuousStar R] :

StarModule M (ContinuousMapZero X R)

Equations

⋯ = ⋯

@[simp]

theorem ContinuousMapZero.coe_star {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] [StarRing R] [ContinuousStar R] (f : ContinuousMapZero X R) :

⇑(star f) = star ⇑f

instance ContinuousMapZero.instTrivialStar {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] [StarRing R] [ContinuousStar R] [TrivialStar R] :

TrivialStar (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instCanLift {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] :

CanLift C(X, R) (ContinuousMapZero X R) toContinuousMap fun (f : C(X, R)) => f 0 = 0

Equations

⋯ = ⋯

def ContinuousMapZero.toContinuousMapHom {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] [StarRing R] [ContinuousStar R] :

ContinuousMapZero X R →⋆ₙₐ[R] C(X, R)

The coercion C(X, R)₀ → C(X, R) bundled as a non-unital star algebra homomorphism.

Equations

ContinuousMapZero.toContinuousMapHom = { toFun := fun (f : ContinuousMapZero X R) => ↑f, map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯, map_mul' := ⋯, map_star' := ⋯ }

Instances For

@[simp]

theorem ContinuousMapZero.toContinuousMapHom_apply {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] [StarRing R] [ContinuousStar R] (f : ContinuousMapZero X R) :

ContinuousMapZero.toContinuousMapHom f = ↑f

theorem ContinuousMapZero.coe_toContinuousMapHom {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] [StarRing R] [ContinuousStar R] :

⇑ContinuousMapZero.toContinuousMapHom = toContinuousMap

def ContinuousMapZero.toContinuousMapCLM {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] (M : Type u_3) [Semiring M] [Module M R] [ContinuousConstSMul M R] :

ContinuousMapZero X R →L[M] C(X, R)

The coercion C(X, R)₀ → C(X, R) bundled as a continuous linear map.

Equations

ContinuousMapZero.toContinuousMapCLM M = { toFun := fun (f : ContinuousMapZero X R) => ↑f, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }

Instances For

@[simp]

theorem ContinuousMapZero.toContinuousMapCLM_apply {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] (M : Type u_3) [Semiring M] [Module M R] [ContinuousConstSMul M R] (f : ContinuousMapZero X R) :

(ContinuousMapZero.toContinuousMapCLM M) f = ↑f

@[simp]

theorem ContinuousMapZero.toContinuousMapCLM_toFun {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] (M : Type u_3) [Semiring M] [Module M R] [ContinuousConstSMul M R] (f : ContinuousMapZero X R) :

(ContinuousMapZero.toContinuousMapCLM M) f = ↑f

def ContinuousMapZero.evalCLM {X : Type u_1} [Zero X] [TopologicalSpace X] (𝕜 : Type u_3) {R : Type u_4} [CompactSpace X] [NormedField 𝕜] [NormedCommRing R] [NormedSpace 𝕜 R] (x : X) :

ContinuousMapZero X R →L[𝕜] R

The evaluation at a point, as a continuous linear map from C(X, R)₀ to R.

Equations

ContinuousMapZero.evalCLM 𝕜 x = (ContinuousMap.evalCLM 𝕜 x).comp (ContinuousMapZero.toContinuousMapCLM 𝕜)

Instances For

@[simp]

theorem ContinuousMapZero.evalCLM_apply {X : Type u_1} [Zero X] [TopologicalSpace X] {𝕜 : Type u_3} {R : Type u_4} [CompactSpace X] [NormedField 𝕜] [NormedCommRing R] [NormedSpace 𝕜 R] (x : X) (f : ContinuousMapZero X R) :

(ContinuousMapZero.evalCLM 𝕜 x) f = f x

def ContinuousMapZero.coeFnAddMonoidHom {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] :

ContinuousMapZero X R →+ X → R

Coercion to a function as an AddMonoidHom. Similar to ContinuousMap.coeFnAddMonoidHom.

Equations

ContinuousMapZero.coeFnAddMonoidHom = { toFun := fun (f : ContinuousMapZero X R) => ⇑f, map_zero' := ⋯, map_add' := ⋯ }

Instances For

@[simp]

theorem ContinuousMapZero.coe_sum {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] {ι : Type u_3} (s : Finset ι) (f : ι → ContinuousMapZero X R) :

⇑(s.sum f) = ∑ i ∈ s, ⇑(f i)

instance ContinuousMapZero.instSub {X : Type u_3} {R : Type u_4} [Zero X] [TopologicalSpace X] [CommRing R] [TopologicalSpace R] [TopologicalRing R] :

Sub (ContinuousMapZero X R)

Equations

ContinuousMapZero.instSub = { sub := fun (f g : ContinuousMapZero X R) => { toContinuousMap := ↑f - ↑g, map_zero' := ⋯ } }

instance ContinuousMapZero.instNeg {X : Type u_3} {R : Type u_4} [Zero X] [TopologicalSpace X] [CommRing R] [TopologicalSpace R] [TopologicalRing R] :

Neg (ContinuousMapZero X R)

Equations

ContinuousMapZero.instNeg = { neg := fun (f : ContinuousMapZero X R) => { toContinuousMap := -↑f, map_zero' := ⋯ } }

instance ContinuousMapZero.instNonUnitalCommRing {X : Type u_3} {R : Type u_4} [Zero X] [TopologicalSpace X] [CommRing R] [TopologicalSpace R] [TopologicalRing R] :

NonUnitalCommRing (ContinuousMapZero X R)

Equations

ContinuousMapZero.instNonUnitalCommRing = Function.Injective.nonUnitalCommRing toContinuousMap ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

@[simp]

theorem ContinuousMapZero.coe_neg {X : Type u_3} {R : Type u_4} [Zero X] [TopologicalSpace X] [CommRing R] [TopologicalSpace R] [TopologicalRing R] (f : ContinuousMapZero X R) :

⇑(-f) = -⇑f

instance ContinuousMapZero.instContinuousNeg {X : Type u_3} {R : Type u_4} [Zero X] [TopologicalSpace X] [CommRing R] [TopologicalSpace R] [TopologicalRing R] :

ContinuousNeg (ContinuousMapZero X R)

Equations

⋯ = ⋯

instance ContinuousMapZero.instUniformSpace {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [Zero R] [UniformSpace R] :

UniformSpace (ContinuousMapZero X R)

Equations

ContinuousMapZero.instUniformSpace = UniformSpace.comap ContinuousMapZero.toContinuousMap inferInstance

theorem ContinuousMapZero.isUniformEmbedding_toContinuousMap {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [Zero R] [UniformSpace R] :

IsUniformEmbedding toContinuousMap

@[deprecated ContinuousMapZero.isUniformEmbedding_toContinuousMap]

theorem ContinuousMapZero.uniformEmbedding_toContinuousMap {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [Zero R] [UniformSpace R] :

IsUniformEmbedding toContinuousMap

Alias of ContinuousMapZero.isUniformEmbedding_toContinuousMap.

instance ContinuousMapZero.instCompleteSpaceOfT1SpaceOfContinuousMap {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [Zero R] [UniformSpace R] [T1Space R] [CompleteSpace C(X, R)] :

CompleteSpace (ContinuousMapZero X R)

Equations

⋯ = ⋯

theorem ContinuousMapZero.isUniformEmbedding_comp {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [Zero R] [UniformSpace R] {Y : Type u_3} [UniformSpace Y] [Zero Y] (g : ContinuousMapZero Y R) (hg : IsUniformEmbedding ⇑g) :

IsUniformEmbedding fun (x : ContinuousMapZero X Y) => g.comp x

@[deprecated ContinuousMapZero.isUniformEmbedding_comp]

theorem ContinuousMapZero.uniformEmbedding_comp {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [Zero R] [UniformSpace R] {Y : Type u_3} [UniformSpace Y] [Zero Y] (g : ContinuousMapZero Y R) (hg : IsUniformEmbedding ⇑g) :

IsUniformEmbedding fun (x : ContinuousMapZero X Y) => g.comp x

Alias of ContinuousMapZero.isUniformEmbedding_comp.

def UniformEquiv.arrowCongrLeft₀ {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [Zero R] [UniformSpace R] {Y : Type u_3} [TopologicalSpace Y] [Zero Y] (f : X ≃ₜ Y) (hf : f 0 = 0) :

ContinuousMapZero X R ≃ᵤ ContinuousMapZero Y R

The uniform equivalence C(X, R)₀ ≃ᵤ C(Y, R)₀ induced by a homeomorphism of the domains sending 0 : X to 0 : Y.

Equations

One or more equations did not get rendered due to their size.

Instances For

def ContinuousMapZero.nonUnitalStarAlgHom_precomp {X : Type u_1} {Y : Type u_2} (R : Type u_4) [Zero X] [Zero Y] [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace R] [CommSemiring R] [StarRing R] [TopologicalSemiring R] [ContinuousStar R] (f : ContinuousMapZero X Y) :

ContinuousMapZero Y R →⋆ₙₐ[R] ContinuousMapZero X R

The functor C(·, R)₀ from topological spaces with zero (and ContinuousMapZero maps) to non-unital star algebras.

Equations

ContinuousMapZero.nonUnitalStarAlgHom_precomp R f = { toFun := fun (g : ContinuousMapZero Y R) => g.comp f, map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯, map_mul' := ⋯, map_star' := ⋯ }

Instances For

@[simp]

theorem ContinuousMapZero.nonUnitalStarAlgHom_precomp_apply {X : Type u_1} {Y : Type u_2} (R : Type u_4) [Zero X] [Zero Y] [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace R] [CommSemiring R] [StarRing R] [TopologicalSemiring R] [ContinuousStar R] (f : ContinuousMapZero X Y) (g : ContinuousMapZero Y R) :

(ContinuousMapZero.nonUnitalStarAlgHom_precomp R f) g = g.comp f

def ContinuousMapZero.nonUnitalStarAlgHom_postcomp (X : Type u_1) {M : Type u_3} {R : Type u_4} {S : Type u_5} [Zero X] [CommSemiring M] [TopologicalSpace X] [TopologicalSpace R] [TopologicalSpace S] [CommSemiring R] [StarRing R] [TopologicalSemiring R] [ContinuousStar R] [CommSemiring S] [StarRing S] [TopologicalSemiring S] [ContinuousStar S] [Module M R] [Module M S] [ContinuousConstSMul M R] [ContinuousConstSMul M S] (φ : R →⋆ₙₐ[M] S) (hφ : Continuous ⇑φ) :

ContinuousMapZero X R →⋆ₙₐ[M] ContinuousMapZero X S

The functor C(X, ·)₀ from non-unital topological star algebras (with non-unital continuous star homomorphisms) to non-unital star algebras.

Equations

One or more equations did not get rendered due to their size.

Instances For

@[simp]

theorem ContinuousMapZero.nonUnitalStarAlgHom_postcomp_apply (X : Type u_1) {M : Type u_3} {R : Type u_4} {S : Type u_5} [Zero X] [CommSemiring M] [TopologicalSpace X] [TopologicalSpace R] [TopologicalSpace S] [CommSemiring R] [StarRing R] [TopologicalSemiring R] [ContinuousStar R] [CommSemiring S] [StarRing S] [TopologicalSemiring S] [ContinuousStar S] [Module M R] [Module M S] [ContinuousConstSMul M R] [ContinuousConstSMul M S] (φ : R →⋆ₙₐ[M] S) (hφ : Continuous ⇑φ) (f : ContinuousMapZero X R) :

(ContinuousMapZero.nonUnitalStarAlgHom_postcomp X φ hφ) f = { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp f

noncomputable instance ContinuousMapZero.instMetricSpace {α : Type u_1} {R : Type u_3} [TopologicalSpace α] [CompactSpace α] [Zero α] [MetricSpace R] [Zero R] :

MetricSpace (ContinuousMapZero α R)

Equations

ContinuousMapZero.instMetricSpace = IsUniformEmbedding.comapMetricSpace toContinuousMap ⋯

noncomputable instance ContinuousMapZero.instNorm {α : Type u_1} {R : Type u_3} [TopologicalSpace α] [CompactSpace α] [Zero α] [NormedAddCommGroup R] :

Norm (ContinuousMapZero α R)

Equations

ContinuousMapZero.instNorm = { norm := fun (f : ContinuousMapZero α R) => ‖↑f‖ }

theorem ContinuousMapZero.norm_def {α : Type u_1} {R : Type u_3} [TopologicalSpace α] [CompactSpace α] [Zero α] [NormedAddCommGroup R] (f : ContinuousMapZero α R) :

‖f‖ = ‖↑f‖

noncomputable instance ContinuousMapZero.instNonUnitalNormedCommRing {α : Type u_1} {R : Type u_3} [TopologicalSpace α] [CompactSpace α] [Zero α] [NormedCommRing R] :

NonUnitalNormedCommRing (ContinuousMapZero α R)

Equations

ContinuousMapZero.instNonUnitalNormedCommRing = NonUnitalNormedCommRing.mk ⋯

instance ContinuousMapZero.instNormedSpaceOfNormedAlgebra {α : Type u_1} {𝕜 : Type u_2} {R : Type u_3} [TopologicalSpace α] [CompactSpace α] [Zero α] [NormedField 𝕜] [NormedCommRing R] [NormedAlgebra 𝕜 R] :

NormedSpace 𝕜 (ContinuousMapZero α R)

Equations

ContinuousMapZero.instNormedSpaceOfNormedAlgebra = NormedSpace.mk ⋯

instance ContinuousMapZero.instCStarRing {α : Type u_1} {R : Type u_3} [TopologicalSpace α] [CompactSpace α] [Zero α] [NormedCommRing R] [StarRing R] [CStarRing R] :

CStarRing (ContinuousMapZero α R)

Equations

⋯ = ⋯