Torsors of additive normed group actions. #

This file defines torsors of additive normed group actions, with a metric space structure. The motivating case is Euclidean affine spaces.

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class NormedAddTorsor (V : outParam (Type u_1)) (P : Type u_2) [SeminormedAddCommGroup V] [PseudoMetricSpace P] extends AddTorsor V P :

Type (max u_1 u_2)

A NormedAddTorsor V P is a torsor of an additive seminormed group action by a SeminormedAddCommGroup V on points P. We bundle the pseudometric space structure and require the distance to be the same as results from the norm (which in fact implies the distance yields a pseudometric space, but bundling just the distance and using an instance for the pseudometric space results in type class problems).

vadd : V → P → P
zero_vadd : ∀ (p : P), 0 +ᵥ p = p
add_vadd : ∀ (g₁ g₂ : V) (p : P), (g₁ + g₂) +ᵥ p = g₁ +ᵥ g₂ +ᵥ p
vsub : P → P → V
nonempty : Nonempty P
vsub_vadd' : ∀ (p₁ p₂ : P), (p₁ -ᵥ p₂) +ᵥ p₂ = p₁
vadd_vsub' : ∀ (g : V) (p : P), (g +ᵥ p) -ᵥ p = g
dist_eq_norm' : ∀ (x y : P), dist x y = ‖x -ᵥ y‖

Instances

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@[instance 100]

instance NormedAddTorsor.toAddTorsor' {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] :

AddTorsor V P

Shortcut instance to help typeclass inference out.

Equations

NormedAddTorsor.toAddTorsor' = NormedAddTorsor.toAddTorsor

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@[instance 100]

instance NormedAddTorsor.to_isometricVAdd {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] :

IsometricVAdd V P

Equations

⋯ = ⋯

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@[instance 100]

instance SeminormedAddCommGroup.toNormedAddTorsor {V : Type u_2} [SeminormedAddCommGroup V] :

NormedAddTorsor V V

A SeminormedAddCommGroup is a NormedAddTorsor over itself.

Equations

SeminormedAddCommGroup.toNormedAddTorsor = NormedAddTorsor.mk ⋯

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instance AffineSubspace.toNormedAddTorsor {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {R : Type u_6} [Ring R] [Module R V] (s : AffineSubspace R P) [Nonempty ↥s] :

NormedAddTorsor ↥s.direction ↥s

A nonempty affine subspace of a NormedAddTorsor is itself a NormedAddTorsor.

Equations

s.toNormedAddTorsor = NormedAddTorsor.mk ⋯

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theorem dist_eq_norm_vsub (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P) :

dist x y = ‖x -ᵥ y‖

The distance equals the norm of subtracting two points. In this lemma, it is necessary to have V as an explicit argument; otherwise rw dist_eq_norm_vsub sometimes doesn't work.

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theorem nndist_eq_nnnorm_vsub (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P) :

nndist x y = ‖x -ᵥ y‖₊

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theorem dist_eq_norm_vsub' (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P) :

dist x y = ‖y -ᵥ x‖

The distance equals the norm of subtracting two points. In this lemma, it is necessary to have V as an explicit argument; otherwise rw dist_eq_norm_vsub' sometimes doesn't work.

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theorem nndist_eq_nnnorm_vsub' (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P) :

nndist x y = ‖y -ᵥ x‖₊

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theorem dist_vadd_cancel_left {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x y : P) :

dist (v +ᵥ x) (v +ᵥ y) = dist x y

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theorem nndist_vadd_cancel_left {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x y : P) :

nndist (v +ᵥ x) (v +ᵥ y) = nndist x y

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@[simp]

theorem dist_vadd_cancel_right {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v₁ v₂ : V) (x : P) :

dist (v₁ +ᵥ x) (v₂ +ᵥ x) = dist v₁ v₂

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@[simp]

theorem nndist_vadd_cancel_right {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v₁ v₂ : V) (x : P) :

nndist (v₁ +ᵥ x) (v₂ +ᵥ x) = nndist v₁ v₂

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@[simp]

theorem dist_vadd_left {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x : P) :

dist (v +ᵥ x) x = ‖v‖

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@[simp]

theorem nndist_vadd_left {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x : P) :

nndist (v +ᵥ x) x = ‖v‖₊

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@[simp]

theorem dist_vadd_right {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x : P) :

dist x (v +ᵥ x) = ‖v‖

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@[simp]

theorem nndist_vadd_right {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x : P) :

nndist x (v +ᵥ x) = ‖v‖₊

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def IsometryEquiv.vaddConst {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) :

V ≃ᵢ P

Isometry between the tangent space V of a (semi)normed add torsor P and P given by addition/subtraction of x : P.

Equations

IsometryEquiv.vaddConst x = { toEquiv := Equiv.vaddConst x, isometry_toFun := ⋯ }

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@[simp]

theorem IsometryEquiv.vaddConst_apply {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (v : V) :

(IsometryEquiv.vaddConst x) v = v +ᵥ x

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@[simp]

theorem IsometryEquiv.vaddConst_symm_apply {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x p' : P) :

(IsometryEquiv.vaddConst x).symm p' = p' -ᵥ x

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@[simp]

theorem IsometryEquiv.vaddConst_toFun {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (v : V) :

(IsometryEquiv.vaddConst x) v = v +ᵥ x

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@[simp]

theorem IsometryEquiv.vaddConst_invFun {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x p' : P) :

(IsometryEquiv.vaddConst x).invFun p' = p' -ᵥ x

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@[simp]

theorem dist_vsub_cancel_left {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y z : P) :

dist (x -ᵥ y) (x -ᵥ z) = dist y z

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@[simp]

theorem nndist_vsub_cancel_left {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y z : P) :

nndist (x -ᵥ y) (x -ᵥ z) = nndist y z

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def IsometryEquiv.constVSub {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) :

P ≃ᵢ V

Isometry between the tangent space V of a (semi)normed add torsor P and P given by subtraction from x : P.

Equations

IsometryEquiv.constVSub x = { toEquiv := Equiv.constVSub x, isometry_toFun := ⋯ }

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@[simp]

theorem IsometryEquiv.constVSub_toFun {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x x✝ : P) :

(IsometryEquiv.constVSub x) x✝ = x -ᵥ x✝

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@[simp]

theorem IsometryEquiv.constVSub_symm_apply {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (x✝ : V) :

(IsometryEquiv.constVSub x).symm x✝ = -x✝ +ᵥ x

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@[simp]

theorem IsometryEquiv.constVSub_apply {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x x✝ : P) :

(IsometryEquiv.constVSub x) x✝ = x -ᵥ x✝

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@[simp]

theorem IsometryEquiv.constVSub_invFun {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (x✝ : V) :

(IsometryEquiv.constVSub x).invFun x✝ = -x✝ +ᵥ x

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@[simp]

theorem dist_vsub_cancel_right {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y z : P) :

dist (x -ᵥ z) (y -ᵥ z) = dist x y

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@[simp]

theorem nndist_vsub_cancel_right {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y z : P) :

nndist (x -ᵥ z) (y -ᵥ z) = nndist x y

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theorem dist_vadd_vadd_le {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v v' : V) (p p' : P) :

dist (v +ᵥ p) (v' +ᵥ p') ≤ dist v v' + dist p p'

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theorem nndist_vadd_vadd_le {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v v' : V) (p p' : P) :

nndist (v +ᵥ p) (v' +ᵥ p') ≤ nndist v v' + nndist p p'

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theorem dist_vsub_vsub_le {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p₁ p₂ p₃ p₄ : P) :

dist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ dist p₁ p₃ + dist p₂ p₄

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theorem nndist_vsub_vsub_le {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p₁ p₂ p₃ p₄ : P) :

nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ nndist p₁ p₃ + nndist p₂ p₄

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theorem edist_vadd_vadd_le {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v v' : V) (p p' : P) :

edist (v +ᵥ p) (v' +ᵥ p') ≤ edist v v' + edist p p'

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theorem edist_vsub_vsub_le {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p₁ p₂ p₃ p₄ : P) :

edist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ edist p₁ p₃ + edist p₂ p₄

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def pseudoMetricSpaceOfNormedAddCommGroupOfAddTorsor (V : Type u_6) (P : Type u_7) [SeminormedAddCommGroup V] [AddTorsor V P] :

PseudoMetricSpace P

The pseudodistance defines a pseudometric space structure on the torsor. This is not an instance because it depends on V to define a MetricSpace P.

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

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def metricSpaceOfNormedAddCommGroupOfAddTorsor (V : Type u_6) (P : Type u_7) [NormedAddCommGroup V] [AddTorsor V P] :

MetricSpace P

The distance defines a metric space structure on the torsor. This is not an instance because it depends on V to define a MetricSpace P.

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metricSpaceOfNormedAddCommGroupOfAddTorsor V P = MetricSpace.mk ⋯

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theorem LipschitzWith.vadd {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [PseudoEMetricSpace α] {f : α → V} {g : α → P} {Kf Kg : NNReal} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) :

LipschitzWith (Kf + Kg) (f +ᵥ g)

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theorem LipschitzWith.vsub {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [PseudoEMetricSpace α] {f g : α → P} {Kf Kg : NNReal} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) :

LipschitzWith (Kf + Kg) (f -ᵥ g)

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theorem uniformContinuous_vadd {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] :

UniformContinuous fun (x : V × P) => x.1 +ᵥ x.2

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theorem uniformContinuous_vsub {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] :

UniformContinuous fun (x : P × P) => x.1 -ᵥ x.2

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@[instance 100]

instance NormedAddTorsor.to_continuousVAdd {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] :

ContinuousVAdd V P

Equations

⋯ = ⋯

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theorem continuous_vsub {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] :

Continuous fun (x : P × P) => x.1 -ᵥ x.2

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theorem Filter.Tendsto.vsub {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {l : Filter α} {f g : α → P} {x y : P} (hf : Filter.Tendsto f l (nhds x)) (hg : Filter.Tendsto g l (nhds y)) :

Filter.Tendsto (f -ᵥ g) l (nhds (x -ᵥ y))

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theorem Continuous.vsub {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [TopologicalSpace α] {f g : α → P} (hf : Continuous f) (hg : Continuous g) :

Continuous (f -ᵥ g)

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theorem ContinuousAt.vsub {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [TopologicalSpace α] {f g : α → P} {x : α} (hf : ContinuousAt f x) (hg : ContinuousAt g x) :

ContinuousAt (f -ᵥ g) x

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theorem ContinuousWithinAt.vsub {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [TopologicalSpace α] {f g : α → P} {x : α} {s : Set α} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :

ContinuousWithinAt (f -ᵥ g) s x

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theorem ContinuousOn.vsub {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [TopologicalSpace α] {f g : α → P} {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) :

ContinuousOn (f -ᵥ g) s

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theorem Filter.Tendsto.lineMap {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {R : Type u_6} [Ring R] [TopologicalSpace R] [Module R V] [ContinuousSMul R V] {l : Filter α} {f₁ f₂ : α → P} {g : α → R} {p₁ p₂ : P} {c : R} (h₁ : Filter.Tendsto f₁ l (nhds p₁)) (h₂ : Filter.Tendsto f₂ l (nhds p₂)) (hg : Filter.Tendsto g l (nhds c)) :

Filter.Tendsto (fun (x : α) => (AffineMap.lineMap (f₁ x) (f₂ x)) (g x)) l (nhds ((AffineMap.lineMap p₁ p₂) c))

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theorem Filter.Tendsto.midpoint {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {R : Type u_6} [Ring R] [TopologicalSpace R] [Module R V] [ContinuousSMul R V] [Invertible 2] {l : Filter α} {f₁ f₂ : α → P} {p₁ p₂ : P} (h₁ : Filter.Tendsto f₁ l (nhds p₁)) (h₂ : Filter.Tendsto f₂ l (nhds p₂)) :

Filter.Tendsto (fun (x : α) => midpoint R (f₁ x) (f₂ x)) l (nhds (midpoint R p₁ p₂))

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theorem IsClosed.vadd_right_of_isCompact {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s : Set V} {t : Set P} (hs : IsClosed s) (ht : IsCompact t) :

IsClosed (s +ᵥ t)

Documentation

Mathlib.Analysis.Normed.Group.AddTorsor

Torsors of additive normed group actions. #