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Mathlib.Topology.ContinuousMap.Compact

Continuous functions on a compact space #

Continuous functions C(α, β) from a compact space α to a metric space β are automatically bounded, and so acquire various structures inherited from α →ᵇ β.

This file transfers these structures, and restates some lemmas characterising these structures.

If you need a lemma which is proved about α →ᵇ β but not for C(α, β) when α is compact, you should restate it here. You can also use ContinuousMap.equivBoundedOfCompact to move functions back and forth.

def ContinuousMap.equivBoundedOfCompact (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] :

C(α, β) ≃ BoundedContinuousFunction α β

When α is compact, the bounded continuous maps α →ᵇ β are equivalent to C(α, β).

Equations

ContinuousMap.equivBoundedOfCompact α β = { toFun := BoundedContinuousFunction.mkOfCompact, invFun := BoundedContinuousFunction.toContinuousMap, left_inv := ⋯, right_inv := ⋯ }

Instances For

@[simp]

theorem ContinuousMap.equivBoundedOfCompact_symm_apply (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] :

⇑(ContinuousMap.equivBoundedOfCompact α β).symm = BoundedContinuousFunction.toContinuousMap

@[simp]

theorem ContinuousMap.equivBoundedOfCompact_apply (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] :

⇑(ContinuousMap.equivBoundedOfCompact α β) = BoundedContinuousFunction.mkOfCompact

theorem ContinuousMap.isUniformInducing_equivBoundedOfCompact (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] :

IsUniformInducing ⇑(ContinuousMap.equivBoundedOfCompact α β)

@[deprecated ContinuousMap.isUniformInducing_equivBoundedOfCompact]

theorem ContinuousMap.uniformInducing_equivBoundedOfCompact (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] :

IsUniformInducing ⇑(ContinuousMap.equivBoundedOfCompact α β)

Alias of ContinuousMap.isUniformInducing_equivBoundedOfCompact.

theorem ContinuousMap.isUniformEmbedding_equivBoundedOfCompact (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] :

IsUniformEmbedding ⇑(ContinuousMap.equivBoundedOfCompact α β)

@[deprecated ContinuousMap.isUniformEmbedding_equivBoundedOfCompact]

theorem ContinuousMap.uniformEmbedding_equivBoundedOfCompact (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] :

IsUniformEmbedding ⇑(ContinuousMap.equivBoundedOfCompact α β)

Alias of ContinuousMap.isUniformEmbedding_equivBoundedOfCompact.

def ContinuousMap.addEquivBoundedOfCompact (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] [AddMonoid β] [LipschitzAdd β] :

C(α, β) ≃+ BoundedContinuousFunction α β

When α is compact, the bounded continuous maps α →ᵇ 𝕜 are additively equivalent to C(α, 𝕜).

Equations

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

Instances For

@[simp]

theorem ContinuousMap.addEquivBoundedOfCompact_symm_apply (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] [AddMonoid β] [LipschitzAdd β] :

⇑(ContinuousMap.addEquivBoundedOfCompact α β).symm = ⇑(BoundedContinuousFunction.toContinuousMapAddHom α β)

@[simp]

theorem ContinuousMap.addEquivBoundedOfCompact_apply (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] [AddMonoid β] [LipschitzAdd β] :

⇑(ContinuousMap.addEquivBoundedOfCompact α β) = BoundedContinuousFunction.mkOfCompact

instance ContinuousMap.instPseudoMetricSpace (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] :

PseudoMetricSpace C(α, β)

Equations

ContinuousMap.instPseudoMetricSpace α β = IsUniformInducing.comapPseudoMetricSpace ⇑(ContinuousMap.equivBoundedOfCompact α β) ⋯

instance ContinuousMap.instMetricSpace (α : Type u_1) [TopologicalSpace α] [CompactSpace α] {β : Type u_4} [MetricSpace β] :

MetricSpace C(α, β)

Equations

ContinuousMap.instMetricSpace α = IsUniformEmbedding.comapMetricSpace ⇑(ContinuousMap.equivBoundedOfCompact α β) ⋯

def ContinuousMap.isometryEquivBoundedOfCompact (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] :

C(α, β) ≃ᵢ BoundedContinuousFunction α β

When α is compact, and β is a metric space, the bounded continuous maps α →ᵇ β are isometric to C(α, β).

Equations

ContinuousMap.isometryEquivBoundedOfCompact α β = { toEquiv := ContinuousMap.equivBoundedOfCompact α β, isometry_toFun := ⋯ }

Instances For

@[simp]

theorem ContinuousMap.isometryEquivBoundedOfCompact_toEquiv (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] :

(ContinuousMap.isometryEquivBoundedOfCompact α β).toEquiv = ContinuousMap.equivBoundedOfCompact α β

@[simp]

theorem ContinuousMap.isometryEquivBoundedOfCompact_apply (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] :

⇑(ContinuousMap.isometryEquivBoundedOfCompact α β) = BoundedContinuousFunction.mkOfCompact

@[simp]

theorem ContinuousMap.isometryEquivBoundedOfCompact_symm_apply (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] :

⇑(ContinuousMap.isometryEquivBoundedOfCompact α β).symm = BoundedContinuousFunction.toContinuousMap

@[simp]

theorem BoundedContinuousFunction.dist_mkOfCompact {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] (f g : C(α, β)) :

dist (BoundedContinuousFunction.mkOfCompact f) (BoundedContinuousFunction.mkOfCompact g) = dist f g

@[simp]

theorem BoundedContinuousFunction.dist_toContinuousMap {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] (f g : BoundedContinuousFunction α β) :

dist f.toContinuousMap g.toContinuousMap = dist f g

theorem ContinuousMap.dist_apply_le_dist {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] {f g : C(α, β)} (x : α) :

dist (f x) (g x) ≤ dist f g

The pointwise distance is controlled by the distance between functions, by definition.

theorem ContinuousMap.dist_le {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] {f g : C(α, β)} {C : ℝ} (C0 : 0 ≤ C) :

dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C

The distance between two functions is controlled by the supremum of the pointwise distances.

theorem ContinuousMap.dist_le_iff_of_nonempty {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] {f g : C(α, β)} {C : ℝ} [Nonempty α] :

dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C

theorem ContinuousMap.dist_lt_iff_of_nonempty {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] {f g : C(α, β)} {C : ℝ} [Nonempty α] :

dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C

theorem ContinuousMap.dist_lt_of_nonempty {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] {f g : C(α, β)} {C : ℝ} [Nonempty α] (w : ∀ (x : α), dist (f x) (g x) < C) :

dist f g < C

theorem ContinuousMap.dist_lt_iff {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] {f g : C(α, β)} {C : ℝ} (C0 : 0 < C) :

dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C

instance ContinuousMap.instBoundedSMul {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] {R : Type u_4} [Zero R] [Zero β] [PseudoMetricSpace R] [SMul R β] [BoundedSMul R β] :

BoundedSMul R C(α, β)

Equations

⋯ = ⋯

instance ContinuousMap.instNorm {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] :

Equations

ContinuousMap.instNorm = { norm := fun (x : C(α, E)) => dist x 0 }

@[simp]

theorem BoundedContinuousFunction.norm_mkOfCompact {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (f : C(α, E)) :

‖BoundedContinuousFunction.mkOfCompact f‖ = ‖f‖

@[simp]

theorem BoundedContinuousFunction.norm_toContinuousMap_eq {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (f : BoundedContinuousFunction α E) :

‖f.toContinuousMap‖ = ‖f‖

instance ContinuousMap.instSeminormedAddCommGroup {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] :

SeminormedAddCommGroup C(α, E)

Equations

ContinuousMap.instSeminormedAddCommGroup = SeminormedAddCommGroup.mk ⋯

instance ContinuousMap.instNormedAddCommGroup {α : Type u_1} [TopologicalSpace α] [CompactSpace α] {E : Type u_4} [NormedAddCommGroup E] :

NormedAddCommGroup C(α, E)

Equations

ContinuousMap.instNormedAddCommGroup = NormedAddCommGroup.mk ⋯

instance ContinuousMap.instNormOneClassOfNonempty {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] [Nonempty α] [One E] [NormOneClass E] :

NormOneClass C(α, E)

Equations

⋯ = ⋯

theorem ContinuousMap.norm_coe_le_norm {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (f : C(α, E)) (x : α) :

‖f x‖ ≤ ‖f‖

theorem ContinuousMap.dist_le_two_norm {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (f : C(α, E)) (x y : α) :

dist (f x) (f y) ≤ 2 * ‖f‖

Distance between the images of any two points is at most twice the norm of the function.

theorem ContinuousMap.norm_le {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (f : C(α, E)) {C : ℝ} (C0 : 0 ≤ C) :

‖f‖ ≤ C ↔ ∀ (x : α), ‖f x‖ ≤ C

The norm of a function is controlled by the supremum of the pointwise norms.

theorem ContinuousMap.norm_le_of_nonempty {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (f : C(α, E)) [Nonempty α] {M : ℝ} :

‖f‖ ≤ M ↔ ∀ (x : α), ‖f x‖ ≤ M

theorem ContinuousMap.norm_lt_iff {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (f : C(α, E)) {M : ℝ} (M0 : 0 < M) :

‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M

theorem ContinuousMap.nnnorm_lt_iff {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (f : C(α, E)) {M : NNReal} (M0 : 0 < M) :

‖f‖₊ < M ↔ ∀ (x : α), ‖f x‖₊ < M

theorem ContinuousMap.norm_lt_iff_of_nonempty {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (f : C(α, E)) [Nonempty α] {M : ℝ} :

‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M

theorem ContinuousMap.nnnorm_lt_iff_of_nonempty {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (f : C(α, E)) [Nonempty α] {M : NNReal} :

‖f‖₊ < M ↔ ∀ (x : α), ‖f x‖₊ < M

theorem ContinuousMap.apply_le_norm {α : Type u_1} [TopologicalSpace α] [CompactSpace α] (f : C(α, ℝ )) (x : α) :

f x ≤ ‖f‖

theorem ContinuousMap.neg_norm_le_apply {α : Type u_1} [TopologicalSpace α] [CompactSpace α] (f : C(α, ℝ )) (x : α) :

-‖f‖ ≤ f x

theorem ContinuousMap.nnnorm_eq_iSup_nnnorm {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (f : C(α, E)) :

‖f‖₊ = ⨆ (x : α), ‖f x‖₊

theorem ContinuousMap.norm_eq_iSup_norm {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (f : C(α, E)) :

‖f‖ = ⨆ (x : α), ‖f x‖

instance ContinuousMap.instCompactSpaceElemCoeCompacts {X : Type u_4} [TopologicalSpace X] (K : TopologicalSpace.Compacts X) :

CompactSpace ↑↑K

Equations

⋯ = ⋯

theorem ContinuousMap.norm_restrict_mono_set {E : Type u_3} [SeminormedAddCommGroup E] {X : Type u_4} [TopologicalSpace X] (f : C(X, E)) {K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) :

‖ContinuousMap.restrict (↑K) f‖ ≤ ‖ContinuousMap.restrict (↑L) f‖

instance ContinuousMap.instNonUnitalSeminormedRing {α : Type u_1} [TopologicalSpace α] [CompactSpace α] {R : Type u_4} [NonUnitalSeminormedRing R] :

NonUnitalSeminormedRing C(α, R)

Equations

ContinuousMap.instNonUnitalSeminormedRing = NonUnitalSeminormedRing.mk ⋯ ⋯

instance ContinuousMap.instNonUnitalSeminormedCommRing {α : Type u_1} [TopologicalSpace α] [CompactSpace α] {R : Type u_4} [NonUnitalSeminormedCommRing R] :

NonUnitalSeminormedCommRing C(α, R)

Equations

ContinuousMap.instNonUnitalSeminormedCommRing = NonUnitalSeminormedCommRing.mk ⋯

instance ContinuousMap.instSeminormedRing {α : Type u_1} [TopologicalSpace α] [CompactSpace α] {R : Type u_4} [SeminormedRing R] :

SeminormedRing C(α, R)

Equations

ContinuousMap.instSeminormedRing = SeminormedRing.mk ⋯ ⋯

instance ContinuousMap.instSeminormedCommRing {α : Type u_1} [TopologicalSpace α] [CompactSpace α] {R : Type u_4} [SeminormedCommRing R] :

SeminormedCommRing C(α, R)

Equations

ContinuousMap.instSeminormedCommRing = SeminormedCommRing.mk ⋯

instance ContinuousMap.instNonUnitalNormedRing {α : Type u_1} [TopologicalSpace α] [CompactSpace α] {R : Type u_4} [NonUnitalNormedRing R] :

NonUnitalNormedRing C(α, R)

Equations

ContinuousMap.instNonUnitalNormedRing = NonUnitalNormedRing.mk ⋯ ⋯

instance ContinuousMap.instNonUnitalNormedCommRing {α : Type u_1} [TopologicalSpace α] [CompactSpace α] {R : Type u_4} [NonUnitalNormedCommRing R] :

NonUnitalNormedCommRing C(α, R)

Equations

ContinuousMap.instNonUnitalNormedCommRing = NonUnitalNormedCommRing.mk ⋯

instance ContinuousMap.instNormedRing {α : Type u_1} [TopologicalSpace α] [CompactSpace α] {R : Type u_4} [NormedRing R] :

NormedRing C(α, R)

Equations

ContinuousMap.instNormedRing = NormedRing.mk ⋯ ⋯

instance ContinuousMap.instNormedCommRing {α : Type u_1} [TopologicalSpace α] [CompactSpace α] {R : Type u_4} [NormedCommRing R] :

NormedCommRing C(α, R)

Equations

ContinuousMap.instNormedCommRing = NormedCommRing.mk ⋯

instance ContinuousMap.normedSpace {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] {𝕜 : Type u_4} [NormedField 𝕜] [NormedSpace 𝕜 E] :

NormedSpace 𝕜 C(α, E)

Equations

ContinuousMap.normedSpace = NormedSpace.mk ⋯

def ContinuousMap.linearIsometryBoundedOfCompact (α : Type u_1) (E : Type u_3) [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (𝕜 : Type u_4) [NormedField 𝕜] [NormedSpace 𝕜 E] :

C(α, E) ≃ₗᵢ[𝕜] BoundedContinuousFunction α E

When α is compact and 𝕜 is a normed field, the 𝕜-algebra of bounded continuous maps α →ᵇ β is 𝕜-linearly isometric to C(α, β).

Equations

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

Instances For

def ContinuousMap.evalCLM {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] (𝕜 : Type u_4) [NormedField 𝕜] [NormedSpace 𝕜 E] (x : α) :

C(α, E) →L[𝕜] E

The evaluation at a point, as a continuous linear map from C(α, 𝕜) to 𝕜.

Equations

ContinuousMap.evalCLM 𝕜 x = (BoundedContinuousFunction.evalCLM 𝕜 x).comp (ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap

Instances For

@[simp]

theorem ContinuousMap.linearIsometryBoundedOfCompact_symm_apply {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] {𝕜 : Type u_4} [NormedField 𝕜] [NormedSpace 𝕜 E] (f : BoundedContinuousFunction α E) :

(ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap

@[simp]

theorem ContinuousMap.linearIsometryBoundedOfCompact_apply_apply {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] {𝕜 : Type u_4} [NormedField 𝕜] [NormedSpace 𝕜 E] (f : C(α, E)) (a : α) :

((ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜) f) a = f a

@[simp]

theorem ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] {𝕜 : Type u_4} [NormedField 𝕜] [NormedSpace 𝕜 E] :

(ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = ContinuousMap.isometryEquivBoundedOfCompact α E

@[simp]

theorem ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] {𝕜 : Type u_4} [NormedField 𝕜] [NormedSpace 𝕜 E] :

↑(ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv = ContinuousMap.addEquivBoundedOfCompact α E

@[simp]

theorem ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [SeminormedAddCommGroup E] {𝕜 : Type u_4} [NormedField 𝕜] [NormedSpace 𝕜 E] :

(ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜).toEquiv = ContinuousMap.equivBoundedOfCompact α E

@[simp]

theorem ContinuousMap.nnnorm_smul_const {α : Type u_1} [TopologicalSpace α] [CompactSpace α] {R : Type u_4} {β : Type u_5} [NormedAddCommGroup β] [NormedDivisionRing R] [Module R β] [BoundedSMul R β] (f : C(α, R)) (b : β) :

‖f • ContinuousMap.const α b‖₊ = ‖f‖₊ * ‖b‖₊

@[simp]

theorem ContinuousMap.norm_smul_const {α : Type u_1} [TopologicalSpace α] [CompactSpace α] {R : Type u_4} {β : Type u_5} [NormedAddCommGroup β] [NormedDivisionRing R] [Module R β] [BoundedSMul R β] (f : C(α, R)) (b : β) :

‖f • ContinuousMap.const α b‖ = ‖f‖ * ‖b‖

instance ContinuousMap.instNormedAlgebra {α : Type u_1} [TopologicalSpace α] [CompactSpace α] {𝕜 : Type u_4} {γ : Type u_5} [NormedField 𝕜] [SeminormedRing γ] [NormedAlgebra 𝕜 γ] :

NormedAlgebra 𝕜 C(α, γ)

Equations

ContinuousMap.instNormedAlgebra = NormedAlgebra.mk ⋯

We now set up some declarations making it convenient to use uniform continuity.

theorem ContinuousMap.uniform_continuity {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [CompactSpace α] [PseudoMetricSpace β] (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :

∃ δ > 0, ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε

def ContinuousMap.modulus {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [CompactSpace α] [PseudoMetricSpace β] (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :

An arbitrarily chosen modulus of uniform continuity for a given function f and ε > 0.

Equations

f.modulus ε h = Classical.choose ⋯

Instances For

theorem ContinuousMap.modulus_pos {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [CompactSpace α] [PseudoMetricSpace β] (f : C(α, β)) {ε : ℝ} {h : 0 < ε} :

0 < f.modulus ε h

theorem ContinuousMap.dist_lt_of_dist_lt_modulus {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [CompactSpace α] [PseudoMetricSpace β] (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α} (w : dist a b < f.modulus ε h) :

dist (f a) (f b) < ε

def ContinuousLinearMap.compLeftContinuousCompact (X : Type u_1) {𝕜 : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace X] [CompactSpace X] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup β] [NormedSpace 𝕜 β] [SeminormedAddCommGroup γ] [NormedSpace 𝕜 γ] (g : β →L[𝕜] γ) :

C(X, β) →L[𝕜] C(X, γ)

Postcomposition of continuous functions into a normed module by a continuous linear map is a continuous linear map. Transferred version of ContinuousLinearMap.compLeftContinuousBounded, upgraded version of ContinuousLinearMap.compLeftContinuous, similar to LinearMap.compLeft.

Equations

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

Instances For

@[simp]

theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (X : Type u_1) {𝕜 : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace X] [CompactSpace X] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup β] [NormedSpace 𝕜 β] [SeminormedAddCommGroup γ] [NormedSpace 𝕜 γ] (g : β →L[𝕜] γ) :

↑(ContinuousLinearMap.compLeftContinuousCompact X g) = ContinuousLinearMap.compLeftContinuous 𝕜 X g

@[simp]

theorem ContinuousLinearMap.compLeftContinuousCompact_apply (X : Type u_1) {𝕜 : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace X] [CompactSpace X] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup β] [NormedSpace 𝕜 β] [SeminormedAddCommGroup γ] [NormedSpace 𝕜 γ] (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :

((ContinuousLinearMap.compLeftContinuousCompact X g) f) x = g (f x)

Local normal convergence #

A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the sum of its sup-norms on any compact subset is summable. This implies convergence in the topology of C(X, E) (i.e. locally uniform convergence).

theorem ContinuousMap.summable_of_locally_summable_norm {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] {E : Type u_2} [NormedAddCommGroup E] [CompleteSpace E] {ι : Type u_3} {F : ι → C(X, E)} (hF : ∀ (K : TopologicalSpace.Compacts X), Summable fun (i : ι) => ‖ContinuousMap.restrict (↑K) (F i)‖) :

Star structures #

In this section, if β is a normed ⋆-group, then so is the space of continuous functions from α to β, by using the star operation pointwise.

Furthermore, if α is compact and β is a C⋆-ring, then C(α, β) is a C⋆-ring.

theorem BoundedContinuousFunction.mkOfCompact_star {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [SeminormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β] [CompactSpace α] (f : C(α, β)) :

BoundedContinuousFunction.mkOfCompact (star f) = star (BoundedContinuousFunction.mkOfCompact f)

instance ContinuousMap.instNormedStarGroup {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [SeminormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β] [CompactSpace α] :

NormedStarGroup C(α, β)

Equations

⋯ = ⋯

instance ContinuousMap.instCStarRing {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [NonUnitalNormedRing β] [StarRing β] [CStarRing β] :

CStarRing C(α, β)

Equations

⋯ = ⋯