File: Mathlib/Analysis/Distribution/SchwartzSpace.lean

full_name: SchwartzMap
code: /-- A function is a Schwartz function if it is smooth and all derivatives decay faster than
  any power of `‖x‖`. -/
structure SchwartzMap where
  toFun : E → F
  smooth' : ContDiff ℝ ⊤ toFun
  decay' : ∀ k n : ℕ, ∃ C : ℝ, ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n toFun x‖ ≤ C
goal: Type (max u_4 u_5)

full_name: SchwartzMap.smooth'
code: /-- A function is a Schwartz function if it is smooth and all derivatives decay faster than
  any power of `‖x‖`. -/
structure SchwartzMap where
  toFun : E → F
  smooth' : ContDiff ℝ ⊤ toFun
  decay' : ∀ k n : ℕ, ∃ C : ℝ, ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n toFun x‖ ≤ C
goal: ContDiff ℝ ⊤ self.toFun

full_name: SchwartzMap.decay'
code: /-- A function is a Schwartz function if it is smooth and all derivatives decay faster than
  any power of `‖x‖`. -/
structure SchwartzMap where
  toFun : E → F
  smooth' : ContDiff ℝ ⊤ toFun
  decay' : ∀ k n : ℕ, ∃ C : ℝ, ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n toFun x‖ ≤ C
goal: ∃ C, ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n self.toFun x‖ ≤ C

full_name: SchwartzMap.toFun
code: /-- A function is a Schwartz function if it is smooth and all derivatives decay faster than
  any power of `‖x‖`. -/
structure SchwartzMap where
  toFun : E → F
  smooth' : ContDiff ℝ ⊤ toFun
  decay' : ∀ k n : ℕ, ∃ C : ℝ, ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n toFun x‖ ≤ C
goal: E → F

full_name: SchwartzMap.«_aux_Mathlib_Analysis_Distribution_SchwartzSpace___macroRules_SchwartzMap_term𝓢(_,_)_1»
code: /-- A function is a Schwartz function if it is smooth and all derivatives decay faster than
  any power of `‖x‖`. -/
scoped[SchwartzMap] notation "𝓢(" E ", " F ")" => SchwartzMap E F
goal: Lean.Macro

full_name: SchwartzMap._aux_Mathlib_Analysis_Distribution_SchwartzSpace___unexpand_SchwartzMap_1
code: /-- A function is a Schwartz function if it is smooth and all derivatives decay faster than
  any power of `‖x‖`. -/
scoped[SchwartzMap] notation "𝓢(" E ", " F ")" => SchwartzMap E F
goal: Lean.PrettyPrinter.Unexpander

full_name: SchwartzMap.instFunLike
code: instance instFunLike : FunLike 𝓢(E, F) E F 
goal: FunLike (SchwartzMap E F) E F

full_name: SchwartzMap.instCoeFun
code: /-- Helper instance for when there's too many metavariables to apply `DFunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun 𝓢(E, F) fun _ => E → F 
goal: CoeFun (SchwartzMap E F) fun x => E → F

full_name: SchwartzMap.decay
code: /-- All derivatives of a Schwartz function are rapidly decaying. -/
theorem decay (f : 𝓢(E, F)) (k n : ℕ) :
    ∃ C : ℝ, 0 < C ∧ ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ C 
goal: ∃ C, 0 < C ∧ ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C

full_name: SchwartzMap.smooth
code: /-- Every Schwartz function is smooth. -/
theorem smooth (f : 𝓢(E, F)) (n : ℕ∞) : ContDiff ℝ n f 
goal: ContDiff ℝ n ⇑f

full_name: SchwartzMap.continuous
code: /-- Every Schwartz function is continuous. -/
@[continuity]
protected theorem continuous (f : 𝓢(E, F)) : Continuous f 
goal: Continuous ⇑f

full_name: SchwartzMap.instContinuousMapClass
code: instance instContinuousMapClass : ContinuousMapClass 𝓢(E, F) E F 
goal: ContinuousMapClass (SchwartzMap E F) E F

full_name: SchwartzMap.differentiable
code: /-- Every Schwartz function is differentiable. -/
protected theorem differentiable (f : 𝓢(E, F)) : Differentiable ℝ f 
goal: Differentiable ℝ ⇑f

full_name: SchwartzMap.differentiableAt
code: /-- Every Schwartz function is differentiable at any point. -/
protected theorem differentiableAt (f : 𝓢(E, F)) {x : E} : DifferentiableAt ℝ f x 
goal: DifferentiableAt ℝ (⇑f) x

full_name: SchwartzMap.ext
code: @[ext]
theorem ext {f g : 𝓢(E, F)} (h : ∀ x, (f : E → F) x = g x) : f = g 
goal: f = g

full_name: SchwartzMap.isBigO_cocompact_zpow_neg_nat
code: /-- Auxiliary lemma, used in proving the more general result `isBigO_cocompact_rpow`. -/
theorem isBigO_cocompact_zpow_neg_nat (k : ℕ) :
    f =O[cocompact E] fun x => ‖x‖ ^ (-k : ℤ) 
goal: ⇑f =O[Filter.cocompact E] fun x => ‖x‖ ^ (-↑k)

full_name: SchwartzMap.isBigO_cocompact_rpow
code: theorem isBigO_cocompact_rpow [ProperSpace E] (s : ℝ) :
    f =O[cocompact E] fun x => ‖x‖ ^ s 
goal: ⇑f =O[Filter.cocompact E] fun x => ‖x‖ ^ s

full_name: SchwartzMap.isBigO_cocompact_zpow
code: theorem isBigO_cocompact_zpow [ProperSpace E] (k : ℤ) :
    f =O[cocompact E] fun x => ‖x‖ ^ k 
goal: ⇑f =O[Filter.cocompact E] fun x => ‖x‖ ^ k

full_name: SchwartzMap.bounds_nonempty
code: theorem bounds_nonempty (k n : ℕ) (f : 𝓢(E, F)) :
    ∃ c : ℝ, c ∈ { c : ℝ | 0 ≤ c ∧ ∀ x : E, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ c } 
goal: ∃ c, c ∈ {c | 0 ≤ c ∧ ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ c}

full_name: SchwartzMap.bounds_bddBelow
code: theorem bounds_bddBelow (k n : ℕ) (f : 𝓢(E, F)) :
    BddBelow { c | 0 ≤ c ∧ ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ c } 
goal: BddBelow {c | 0 ≤ c ∧ ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ c}

full_name: SchwartzMap.decay_add_le_aux
code: theorem decay_add_le_aux (k n : ℕ) (f g : 𝓢(E, F)) (x : E) :
    ‖x‖ ^ k * ‖iteratedFDeriv ℝ n ((f : E → F) + (g : E → F)) x‖ ≤
      ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ + ‖x‖ ^ k * ‖iteratedFDeriv ℝ n g x‖ 
goal: ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f + ⇑g) x‖ ≤
  ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ + ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑g) x‖

full_name: SchwartzMap.decay_neg_aux
code: theorem decay_neg_aux (k n : ℕ) (f : 𝓢(E, F)) (x : E) :
    ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (-f : E → F) x‖ = ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ 
goal: ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (-⇑f) x‖ = ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖

full_name: SchwartzMap.decay_smul_aux
code: theorem decay_smul_aux (k n : ℕ) (f : 𝓢(E, F)) (c : 𝕜) (x : E) :
    ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (c • (f : E → F)) x‖ =
      ‖c‖ * ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ 
goal: ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (c • ⇑f) x‖ = ‖c‖ * ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖

full_name: SchwartzMap.seminormAux
code: /-- Helper definition for the seminorms of the Schwartz space. -/
protected def seminormAux (k n : ℕ) (f : 𝓢(E, F)) : ℝ 
goal: ℝ

full_name: SchwartzMap.seminormAux_nonneg
code: theorem seminormAux_nonneg (k n : ℕ) (f : 𝓢(E, F)) : 0 ≤ f.seminormAux k n 
goal: 0 ≤ SchwartzMap.seminormAux k n f

full_name: SchwartzMap.le_seminormAux
code: theorem le_seminormAux (k n : ℕ) (f : 𝓢(E, F)) (x : E) :
    ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ f.seminormAux k n 
goal: ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ SchwartzMap.seminormAux k n f

full_name: SchwartzMap.seminormAux_le_bound
code: /-- If one controls the norm of every `A x`, then one controls the norm of `A`. -/
theorem seminormAux_le_bound (k n : ℕ) (f : 𝓢(E, F)) {M : ℝ} (hMp : 0 ≤ M)
    (hM : ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ M) : f.seminormAux k n ≤ M 
goal: SchwartzMap.seminormAux k n f ≤ M

full_name: SchwartzMap.instSMul
code: instance instSMul : SMul 𝕜 𝓢(E, F) 
goal: SMul 𝕜 (SchwartzMap E F)

full_name: SchwartzMap.smul_apply
code: @[simp]
theorem smul_apply {f : 𝓢(E, F)} {c : 𝕜} {x : E} : (c • f) x = c • f x 
goal: (c • f) x = c • f x

full_name: SchwartzMap.instIsScalarTower
code: instance instIsScalarTower [SMul 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' F] : IsScalarTower 𝕜 𝕜' 𝓢(E, F) 
goal: IsScalarTower 𝕜 𝕜' (SchwartzMap E F)

full_name: SchwartzMap.instSMulCommClass
code: instance instSMulCommClass [SMulCommClass 𝕜 𝕜' F] : SMulCommClass 𝕜 𝕜' 𝓢(E, F) 
goal: SMulCommClass 𝕜 𝕜' (SchwartzMap E F)

full_name: SchwartzMap.seminormAux_smul_le
code: theorem seminormAux_smul_le (k n : ℕ) (c : 𝕜) (f : 𝓢(E, F)) :
    (c • f).seminormAux k n ≤ ‖c‖ * f.seminormAux k n 
goal: SchwartzMap.seminormAux k n (c • f) ≤ ‖c‖ * SchwartzMap.seminormAux k n f

full_name: SchwartzMap.instNSMul
code: instance instNSMul : SMul ℕ 𝓢(E, F) 
goal: SMul ℕ (SchwartzMap E F)

full_name: SchwartzMap.instZSMul
code: instance instZSMul : SMul ℤ 𝓢(E, F) 
goal: SMul ℤ (SchwartzMap E F)

full_name: SchwartzMap.instZero
code: instance instZero : Zero 𝓢(E, F) 
goal: Zero (SchwartzMap E F)

full_name: SchwartzMap.instInhabited
code: instance instInhabited : Inhabited 𝓢(E, F) 
goal: Inhabited (SchwartzMap E F)

full_name: SchwartzMap.coe_zero
code: theorem coe_zero : DFunLike.coe (0 : 𝓢(E, F)) = (0 : E → F) 
goal: ⇑0 = 0

full_name: SchwartzMap.coeFn_zero
code: @[simp]
theorem coeFn_zero : ⇑(0 : 𝓢(E, F)) = (0 : E → F) 
goal: ⇑0 = 0

full_name: SchwartzMap.zero_apply
code: @[simp]
theorem zero_apply {x : E} : (0 : 𝓢(E, F)) x = 0 
goal: 0 x = 0

full_name: SchwartzMap.seminormAux_zero
code: theorem seminormAux_zero (k n : ℕ) : (0 : 𝓢(E, F)).seminormAux k n = 0 
goal: SchwartzMap.seminormAux k n 0 = 0

full_name: SchwartzMap.instNeg
code: instance instNeg : Neg 𝓢(E, F) 
goal: Neg (SchwartzMap E F)

full_name: SchwartzMap.instAdd
code: instance instAdd : Add 𝓢(E, F) 
goal: Add (SchwartzMap E F)

full_name: SchwartzMap.add_apply
code: @[simp]
theorem add_apply {f g : 𝓢(E, F)} {x : E} : (f + g) x = f x + g x 
goal: (f + g) x = f x + g x

full_name: SchwartzMap.seminormAux_add_le
code: theorem seminormAux_add_le (k n : ℕ) (f g : 𝓢(E, F)) :
    (f + g).seminormAux k n ≤ f.seminormAux k n + g.seminormAux k n 
goal: SchwartzMap.seminormAux k n (f + g) ≤ SchwartzMap.seminormAux k n f + SchwartzMap.seminormAux k n g

full_name: SchwartzMap.instSub
code: instance instSub : Sub 𝓢(E, F) 
goal: Sub (SchwartzMap E F)

full_name: SchwartzMap.sub_apply
code: @[simp]
theorem sub_apply {f g : 𝓢(E, F)} {x : E} : (f - g) x = f x - g x 
goal: (f - g) x = f x - g x

full_name: SchwartzMap.instAddCommGroup
code: instance instAddCommGroup : AddCommGroup 𝓢(E, F) 
goal: AddCommGroup (SchwartzMap E F)

full_name: SchwartzMap.coeHom
code: /-- Coercion as an additive homomorphism. -/
def coeHom : 𝓢(E, F) →+ E → F 
goal: SchwartzMap E F →+ E → F

full_name: SchwartzMap.coe_coeHom
code: theorem coe_coeHom : (coeHom E F : 𝓢(E, F) → E → F) = DFunLike.coe 
goal: ⇑(SchwartzMap.coeHom E F) = DFunLike.coe

full_name: SchwartzMap.coeHom_injective
code: theorem coeHom_injective : Function.Injective (coeHom E F) 
goal: Function.Injective ⇑(SchwartzMap.coeHom E F)

full_name: SchwartzMap.instModule
code: instance instModule : Module 𝕜 𝓢(E, F) 
goal: Module 𝕜 (SchwartzMap E F)

full_name: SchwartzMap.seminorm
code: /-- The seminorms of the Schwartz space given by the best constants in the definition of
`𝓢(E, F)`. -/
protected def seminorm (k n : ℕ) : Seminorm 𝕜 𝓢(E, F) 
goal: Seminorm 𝕜 (SchwartzMap E F)

full_name: SchwartzMap.seminorm_le_bound
code: /-- If one controls the seminorm for every `x`, then one controls the seminorm. -/
theorem seminorm_le_bound (k n : ℕ) (f : 𝓢(E, F)) {M : ℝ} (hMp : 0 ≤ M)
    (hM : ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ M) : SchwartzMap.seminorm 𝕜 k n f ≤ M 
goal: (SchwartzMap.seminorm 𝕜 k n) f ≤ M

full_name: SchwartzMap.seminorm_le_bound'
code: /-- If one controls the seminorm for every `x`, then one controls the seminorm.

Variant for functions `𝓢(ℝ, F)`. -/
theorem seminorm_le_bound' (k n : ℕ) (f : 𝓢(ℝ, F)) {M : ℝ} (hMp : 0 ≤ M)
    (hM : ∀ x, |x| ^ k * ‖iteratedDeriv n f x‖ ≤ M) : SchwartzMap.seminorm 𝕜 k n f ≤ M 
goal: (SchwartzMap.seminorm 𝕜 k n) f ≤ M

full_name: SchwartzMap.le_seminorm
code: /-- The seminorm controls the Schwartz estimate for any fixed `x`. -/
theorem le_seminorm (k n : ℕ) (f : 𝓢(E, F)) (x : E) :
    ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ SchwartzMap.seminorm 𝕜 k n f 
goal: ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ (SchwartzMap.seminorm 𝕜 k n) f

full_name: SchwartzMap.le_seminorm'
code: /-- The seminorm controls the Schwartz estimate for any fixed `x`.

Variant for functions `𝓢(ℝ, F)`. -/
theorem le_seminorm' (k n : ℕ) (f : 𝓢(ℝ, F)) (x : ℝ) :
    |x| ^ k * ‖iteratedDeriv n f x‖ ≤ SchwartzMap.seminorm 𝕜 k n f 
goal: |x| ^ k * ‖iteratedDeriv n (⇑f) x‖ ≤ (SchwartzMap.seminorm 𝕜 k n) f

full_name: SchwartzMap.norm_iteratedFDeriv_le_seminorm
code: theorem norm_iteratedFDeriv_le_seminorm (f : 𝓢(E, F)) (n : ℕ) (x₀ : E) :
    ‖iteratedFDeriv ℝ n f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 0 n) f 
goal: ‖iteratedFDeriv ℝ n (⇑f) x₀‖ ≤ (SchwartzMap.seminorm 𝕜 0 n) f

full_name: SchwartzMap.norm_pow_mul_le_seminorm
code: theorem norm_pow_mul_le_seminorm (f : 𝓢(E, F)) (k : ℕ) (x₀ : E) :
    ‖x₀‖ ^ k * ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 k 0) f 
goal: ‖x₀‖ ^ k * ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 k 0) f

full_name: SchwartzMap.norm_le_seminorm
code: theorem norm_le_seminorm (f : 𝓢(E, F)) (x₀ : E) : ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 0 0) f 
goal: ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 0 0) f

full_name: schwartzSeminormFamily
code: /-- The family of Schwartz seminorms. -/
def _root_.schwartzSeminormFamily : SeminormFamily 𝕜 𝓢(E, F) (ℕ × ℕ) 
goal: SeminormFamily 𝕜 (SchwartzMap E F) (ℕ × ℕ)

full_name: SchwartzMap.schwartzSeminormFamily_apply
code: @[simp]
theorem schwartzSeminormFamily_apply (n k : ℕ) :
    schwartzSeminormFamily 𝕜 E F (n, k) = SchwartzMap.seminorm 𝕜 n k 
goal: schwartzSeminormFamily 𝕜 E F (n, k) = SchwartzMap.seminorm 𝕜 n k

full_name: SchwartzMap.schwartzSeminormFamily_apply_zero
code: @[simp]
theorem schwartzSeminormFamily_apply_zero :
    schwartzSeminormFamily 𝕜 E F 0 = SchwartzMap.seminorm 𝕜 0 0 
goal: schwartzSeminormFamily 𝕜 E F 0 = SchwartzMap.seminorm 𝕜 0 0

full_name: SchwartzMap.one_add_le_sup_seminorm_apply
code: /-- A more convenient version of `le_sup_seminorm_apply`.

The set `Finset.Iic m` is the set of all pairs `(k', n')` with `k' ≤ m.1` and `n' ≤ m.2`.
Note that the constant is far from optimal. -/
theorem one_add_le_sup_seminorm_apply {m : ℕ × ℕ} {k n : ℕ} (hk : k ≤ m.1) (hn : n ≤ m.2)
    (f : 𝓢(E, F)) (x : E) :
    (1 + ‖x‖) ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤
      2 ^ m.1 * (Finset.Iic m).sup (fun m => SchwartzMap.seminorm 𝕜 m.1 m.2) f 
goal: (1 + ‖x‖) ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ 2 ^ m.1 * ((Finset.Iic m).sup fun m => SchwartzMap.seminorm 𝕜 m.1 m.2) f

full_name: SchwartzMap.instTopologicalSpace
code: instance instTopologicalSpace : TopologicalSpace 𝓢(E, F) 
goal: TopologicalSpace (SchwartzMap E F)

full_name: schwartz_withSeminorms
code: theorem _root_.schwartz_withSeminorms : WithSeminorms (schwartzSeminormFamily 𝕜 E F) 
goal: WithSeminorms (schwartzSeminormFamily 𝕜 E F)

full_name: SchwartzMap.instContinuousSMul
code: instance instContinuousSMul : ContinuousSMul 𝕜 𝓢(E, F) 
goal: ContinuousSMul 𝕜 (SchwartzMap E F)

full_name: SchwartzMap.instTopologicalAddGroup
code: instance instTopologicalAddGroup : TopologicalAddGroup 𝓢(E, F) 
goal: TopologicalAddGroup (SchwartzMap E F)

full_name: SchwartzMap.instUniformSpace
code: instance instUniformSpace : UniformSpace 𝓢(E, F) 
goal: UniformSpace (SchwartzMap E F)

full_name: SchwartzMap.instUniformAddGroup
code: instance instUniformAddGroup : UniformAddGroup 𝓢(E, F) 
goal: UniformAddGroup (SchwartzMap E F)

full_name: SchwartzMap.instLocallyConvexSpace
code: instance instLocallyConvexSpace : LocallyConvexSpace ℝ 𝓢(E, F) 
goal: LocallyConvexSpace ℝ (SchwartzMap E F)

full_name: SchwartzMap.instFirstCountableTopology
code: instance instFirstCountableTopology : FirstCountableTopology 𝓢(E, F) 
goal: FirstCountableTopology (SchwartzMap E F)

full_name: Function.HasTemperateGrowth
code: /-- A function is called of temperate growth if it is smooth and all iterated derivatives are
polynomially bounded. -/
def _root_.Function.HasTemperateGrowth (f : E → F) : Prop 
goal: Prop

full_name: Function.HasTemperateGrowth.norm_iteratedFDeriv_le_uniform_aux
code: theorem _root_.Function.HasTemperateGrowth.norm_iteratedFDeriv_le_uniform_aux {f : E → F}
    (hf_temperate : f.HasTemperateGrowth) (n : ℕ) :
    ∃ (k : ℕ) (C : ℝ), 0 ≤ C ∧ ∀ N ≤ n, ∀ x : E, ‖iteratedFDeriv ℝ N f x‖ ≤ C * (1 + ‖x‖) ^ k 
goal: ∃ k C, 0 ≤ C ∧ ∀ N ≤ n, ∀ (x : E), ‖iteratedFDeriv ℝ N f x‖ ≤ C * (1 + ‖x‖) ^ k

full_name: Function.HasTemperateGrowth.of_fderiv
code: lemma _root_.Function.HasTemperateGrowth.of_fderiv {f : E → F}
    (h'f : Function.HasTemperateGrowth (fderiv ℝ f)) (hf : Differentiable ℝ f) {k : ℕ} {C : ℝ}
    (h : ∀ x, ‖f x‖ ≤ C * (1 + ‖x‖) ^ k) :
    Function.HasTemperateGrowth f 
goal: Function.HasTemperateGrowth f

full_name: Function.HasTemperateGrowth.zero
code: lemma _root_.Function.HasTemperateGrowth.zero :
    Function.HasTemperateGrowth (fun _ : E ↦ (0 : F)) 
goal: Function.HasTemperateGrowth fun x => 0

full_name: Function.HasTemperateGrowth.const
code: lemma _root_.Function.HasTemperateGrowth.const (c : F) :
    Function.HasTemperateGrowth (fun _ : E ↦ c) 
goal: Function.HasTemperateGrowth fun x => c

full_name: ContinuousLinearMap.hasTemperateGrowth
code: lemma _root_.ContinuousLinearMap.hasTemperateGrowth (f : E →L[ℝ] F) :
    Function.HasTemperateGrowth f 
goal: Function.HasTemperateGrowth ⇑f

full_name: MeasureTheory.Measure.HasTemperateGrowth.exists_integrable
code: /-- A measure `μ` has temperate growth if there is an `n : ℕ` such that `(1 + ‖x‖) ^ (- n)` is
`μ`-integrable. -/
class _root_.MeasureTheory.Measure.HasTemperateGrowth (μ : Measure D) : Prop :=
  exists_integrable : ∃ (n : ℕ), Integrable (fun x ↦ (1 + ‖x‖) ^ (- (n : ℝ))) μ
goal: ∃ n, MeasureTheory.Integrable (fun x => (1 + ‖x‖) ^ (-↑n)) μ

full_name: MeasureTheory.Measure.HasTemperateGrowth
code: /-- A measure `μ` has temperate growth if there is an `n : ℕ` such that `(1 + ‖x‖) ^ (- n)` is
`μ`-integrable. -/
class _root_.MeasureTheory.Measure.HasTemperateGrowth (μ : Measure D) : Prop :=
  exists_integrable : ∃ (n : ℕ), Integrable (fun x ↦ (1 + ‖x‖) ^ (- (n : ℝ))) μ
goal: Prop

full_name: MeasureTheory.Measure.integrablePower
code: /-- An integer exponent `l` such that `(1 + ‖x‖) ^ (-l)` is integrable if `μ` has
temperate growth. -/
def _root_.MeasureTheory.Measure.integrablePower (μ : Measure D) : ℕ 
goal: ℕ

full_name: SchwartzMap.integrable_pow_neg_integrablePower
code: lemma integrable_pow_neg_integrablePower
    (μ : Measure D) [h : μ.HasTemperateGrowth] :
    Integrable (fun x ↦ (1 + ‖x‖) ^ (- (μ.integrablePower : ℝ))) μ 
goal: MeasureTheory.Integrable (fun x => (1 + ‖x‖) ^ (-↑μ.integrablePower)) μ

full_name: MeasureTheory.Measure.IsFiniteMeasure.instHasTemperateGrowth
code: instance _root_.MeasureTheory.Measure.IsFiniteMeasure.instHasTemperateGrowth {μ : Measure D}
    [h : IsFiniteMeasure μ] : μ.HasTemperateGrowth 
goal: μ.HasTemperateGrowth

full_name: MeasureTheory.Measure.IsAddHaarMeasure.instHasTemperateGrowth
code: instance _root_.MeasureTheory.Measure.IsAddHaarMeasure.instHasTemperateGrowth {μ : Measure D}
    [h : μ.IsAddHaarMeasure] : μ.HasTemperateGrowth 
goal: μ.HasTemperateGrowth

full_name: SchwartzMap.pow_mul_le_of_le_of_pow_mul_le
code: /-- Pointwise inequality to control `x ^ k * f` in terms of `1 / (1 + x) ^ l` if one controls both
`f` (with a bound `C₁`) and `x ^ (k + l) * f` (with a bound `C₂`). This will be used to check
integrability of `x ^ k * f x` when `f` is a Schwartz function, and to control explicitly its
integral in terms of suitable seminorms of `f`. -/
lemma pow_mul_le_of_le_of_pow_mul_le {C₁ C₂ : ℝ} {k l : ℕ} {x f : ℝ} (hx : 0 ≤ x) (hf : 0 ≤ f)
    (h₁ : f ≤ C₁) (h₂ : x ^ (k + l) * f ≤ C₂) :
    x ^ k * f ≤ 2 ^ l * (C₁ + C₂) * (1 + x) ^ (- (l : ℝ)) 
goal: x ^ k * f ≤ 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l)

full_name: SchwartzMap.integrable_of_le_of_pow_mul_le
code: /-- Given a function such that `f` and `x ^ (k + l) * f` are bounded for a suitable `l`, then
`x ^ k * f` is integrable. The bounds are not relevant for the integrability conclusion, but they
are relevant for bounding the integral in `integral_pow_mul_le_of_le_of_pow_mul_le`. We formulate
the two lemmas with the same set of assumptions for ease of applications. -/
lemma integrable_of_le_of_pow_mul_le
    {μ : Measure D} [μ.HasTemperateGrowth] {f : D → E} {C₁ C₂ : ℝ} {k : ℕ}
    (hf : ∀ x, ‖f x‖ ≤ C₁) (h'f : ∀ x, ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂)
    (h''f : AEStronglyMeasurable f μ) :
    Integrable (fun x ↦ ‖x‖ ^ k * ‖f x‖) μ 
goal: MeasureTheory.Integrable (fun x => ‖x‖ ^ k * ‖f x‖) μ

full_name: SchwartzMap.integral_pow_mul_le_of_le_of_pow_mul_le
code: /-- Given a function such that `f` and `x ^ (k + l) * f` are bounded for a suitable `l`, then
one can bound explicitly the integral of `x ^ k * f`. -/
lemma integral_pow_mul_le_of_le_of_pow_mul_le
    {μ : Measure D} [μ.HasTemperateGrowth] {f : D → E} {C₁ C₂ : ℝ} {k : ℕ}
    (hf : ∀ x, ‖f x‖ ≤ C₁) (h'f : ∀ x, ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂) :
    ∫ x, ‖x‖ ^ k * ‖f x‖ ∂μ ≤ 2 ^ μ.integrablePower *
      (∫ x, (1 + ‖x‖) ^ (- (μ.integrablePower : ℝ)) ∂μ) * (C₁ + C₂) 
goal: ∫ (x : D), ‖x‖ ^ k * ‖f x‖ ∂μ ≤ (2 ^ μ.integrablePower * ∫ (x : D), (1 + ‖x‖) ^ (-↑μ.integrablePower) ∂μ) * (C₁ + C₂)

full_name: SchwartzMap.mkLM
code: /-- Create a semilinear map between Schwartz spaces.

Note: This is a helper definition for `mkCLM`. -/
def mkLM (A : (D → E) → F → G) (hadd : ∀ (f g : 𝓢(D, E)) (x), A (f + g) x = A f x + A g x)
    (hsmul : ∀ (a : 𝕜) (f : 𝓢(D, E)) (x), A (a • f) x = σ a • A f x)
    (hsmooth : ∀ f : 𝓢(D, E), ContDiff ℝ ⊤ (A f))
    (hbound : ∀ n : ℕ × ℕ, ∃ (s : Finset (ℕ × ℕ)) (C : ℝ), 0 ≤ C ∧ ∀ (f : 𝓢(D, E)) (x : F),
      ‖x‖ ^ n.fst * ‖iteratedFDeriv ℝ n.snd (A f) x‖ ≤ C * s.sup (schwartzSeminormFamily 𝕜 D E) f) :
    𝓢(D, E) →ₛₗ[σ] 𝓢(F, G) 
goal: SchwartzMap D E →ₛₗ[σ] SchwartzMap F G

full_name: SchwartzMap.mkCLM
code: /-- Create a continuous semilinear map between Schwartz spaces.

For an example of using this definition, see `fderivCLM`. -/
def mkCLM [RingHomIsometric σ] (A : (D → E) → F → G)
    (hadd : ∀ (f g : 𝓢(D, E)) (x), A (f + g) x = A f x + A g x)
    (hsmul : ∀ (a : 𝕜) (f : 𝓢(D, E)) (x), A (a • f) x = σ a • A f x)
    (hsmooth : ∀ f : 𝓢(D, E), ContDiff ℝ ⊤ (A f))
    (hbound : ∀ n : ℕ × ℕ, ∃ (s : Finset (ℕ × ℕ)) (C : ℝ), 0 ≤ C ∧ ∀ (f : 𝓢(D, E)) (x : F),
      ‖x‖ ^ n.fst * ‖iteratedFDeriv ℝ n.snd (A f) x‖ ≤ C * s.sup (schwartzSeminormFamily 𝕜 D E) f) :
    𝓢(D, E) →SL[σ] 𝓢(F, G) 
goal: SchwartzMap D E →SL[σ] SchwartzMap F G

full_name: SchwartzMap.mkCLMtoNormedSpace
code: /-- Define a continuous semilinear map from Schwartz space to a normed space. -/
def mkCLMtoNormedSpace [RingHomIsometric σ] (A : 𝓢(D, E) → G)
    (hadd : ∀ (f g : 𝓢(D, E)), A (f + g) = A f + A g)
    (hsmul : ∀ (a : 𝕜) (f : 𝓢(D, E)), A (a • f) = σ a • A f)
    (hbound : ∃ (s : Finset (ℕ × ℕ)) (C : ℝ), 0 ≤ C ∧ ∀ (f : 𝓢(D, E)),
      ‖A f‖ ≤ C * s.sup (schwartzSeminormFamily 𝕜 D E) f) :
    𝓢(D, E) →SL[σ] G 
goal: SchwartzMap D E →SL[σ] G

full_name: SchwartzMap.evalCLM
code: /-- The map applying a vector to Hom-valued Schwartz function as a continuous linear map. -/
protected def evalCLM (m : E) : 𝓢(E, E →L[ℝ] F) →L[𝕜] 𝓢(E, F) 
goal: SchwartzMap E (E →L[ℝ] F) →L[𝕜] SchwartzMap E F

full_name: SchwartzMap.bilinLeftCLM
code: /-- The map `f ↦ (x ↦ B (f x) (g x))` as a continuous `𝕜`-linear map on Schwartz space,
where `B` is a continuous `𝕜`-linear map and `g` is a function of temperate growth. -/
def bilinLeftCLM (B : E →L[ℝ] F →L[ℝ] G) {g : D → F} (hg : g.HasTemperateGrowth) :
    𝓢(D, E) →L[ℝ] 𝓢(D, G) 
goal: SchwartzMap D E →L[ℝ] SchwartzMap D G

full_name: SchwartzMap.compCLM
code: /-- Composition with a function on the right is a continuous linear map on Schwartz space
provided that the function is temperate and growths polynomially near infinity. -/
def compCLM {g : D → E} (hg : g.HasTemperateGrowth)
    (hg_upper : ∃ (k : ℕ) (C : ℝ), ∀ x, ‖x‖ ≤ C * (1 + ‖g x‖) ^ k) : 𝓢(E, F) →L[𝕜] 𝓢(D, F) 
goal: SchwartzMap E F →L[𝕜] SchwartzMap D F

full_name: SchwartzMap.compCLM_apply
code: @[simp] lemma compCLM_apply {g : D → E} (hg : g.HasTemperateGrowth)
    (hg_upper : ∃ (k : ℕ) (C : ℝ), ∀ x, ‖x‖ ≤ C * (1 + ‖g x‖) ^ k) (f : 𝓢(E, F)) :
    compCLM 𝕜 hg hg_upper f = f ∘ g 
goal: ⇑((SchwartzMap.compCLM 𝕜 hg hg_upper) f) = ⇑f ∘ g

full_name: SchwartzMap.compCLMOfAntilipschitz
code: /-- Composition with a function on the right is a continuous linear map on Schwartz space
provided that the function is temperate and antilipschitz. -/
def compCLMOfAntilipschitz {K : ℝ≥0} {g : D → E}
    (hg : g.HasTemperateGrowth) (h'g : AntilipschitzWith K g) :
    𝓢(E, F) →L[𝕜] 𝓢(D, F) 
goal: SchwartzMap E F →L[𝕜] SchwartzMap D F

full_name: SchwartzMap.compCLMOfAntilipschitz_apply
code: @[simp] lemma compCLMOfAntilipschitz_apply {K : ℝ≥0} {g : D → E} (hg : g.HasTemperateGrowth)
    (h'g : AntilipschitzWith K g) (f : 𝓢(E, F)) :
    compCLMOfAntilipschitz 𝕜 hg h'g f = f ∘ g 
goal: ⇑((SchwartzMap.compCLMOfAntilipschitz 𝕜 hg h'g) f) = ⇑f ∘ g

full_name: SchwartzMap.compCLMOfContinuousLinearEquiv
code: /-- Composition with a continuous linear equiv on the right is a continuous linear map on
Schwartz space. -/
def compCLMOfContinuousLinearEquiv (g : D ≃L[ℝ] E) :
    𝓢(E, F) →L[𝕜] 𝓢(D, F) 
goal: SchwartzMap E F →L[𝕜] SchwartzMap D F

full_name: SchwartzMap.compCLMOfContinuousLinearEquiv_apply
code: @[simp] lemma compCLMOfContinuousLinearEquiv_apply (g : D ≃L[ℝ] E) (f : 𝓢(E, F)) :
    compCLMOfContinuousLinearEquiv 𝕜 g f = f ∘ g 
goal: ⇑((SchwartzMap.compCLMOfContinuousLinearEquiv 𝕜 g) f) = ⇑f ∘ ⇑g

full_name: SchwartzMap.fderivCLM
code: /-- The Fréchet derivative on Schwartz space as a continuous `𝕜`-linear map. -/
def fderivCLM : 𝓢(E, F) →L[𝕜] 𝓢(E, E →L[ℝ] F) 
goal: SchwartzMap E F →L[𝕜] SchwartzMap E (E →L[ℝ] F)

full_name: SchwartzMap.fderivCLM_apply
code: @[simp]
theorem fderivCLM_apply (f : 𝓢(E, F)) (x : E) : fderivCLM 𝕜 f x = fderiv ℝ f x 
goal: ((SchwartzMap.fderivCLM 𝕜) f) x = fderiv ℝ (⇑f) x

full_name: SchwartzMap.derivCLM
code: /-- The 1-dimensional derivative on Schwartz space as a continuous `𝕜`-linear map. -/
def derivCLM : 𝓢(ℝ, F) →L[𝕜] 𝓢(ℝ, F) 
goal: SchwartzMap ℝ F →L[𝕜] SchwartzMap ℝ F

full_name: SchwartzMap.derivCLM_apply
code: @[simp]
theorem derivCLM_apply (f : 𝓢(ℝ, F)) (x : ℝ) : derivCLM 𝕜 f x = deriv f x 
goal: ((SchwartzMap.derivCLM 𝕜) f) x = deriv (⇑f) x

full_name: SchwartzMap.pderivCLM
code: /-- The partial derivative (or directional derivative) in the direction `m : E` as a
continuous linear map on Schwartz space. -/
def pderivCLM (m : E) : 𝓢(E, F) →L[𝕜] 𝓢(E, F) 
goal: SchwartzMap E F →L[𝕜] SchwartzMap E F

full_name: SchwartzMap.pderivCLM_apply
code: @[simp]
theorem pderivCLM_apply (m : E) (f : 𝓢(E, F)) (x : E) : pderivCLM 𝕜 m f x = fderiv ℝ f x m 
goal: ((SchwartzMap.pderivCLM 𝕜 m) f) x = (fderiv ℝ (⇑f) x) m

full_name: SchwartzMap.pderivCLM_eq_lineDeriv
code: theorem pderivCLM_eq_lineDeriv (m : E) (f : 𝓢(E, F)) (x : E) :
    pderivCLM 𝕜 m f x = lineDeriv ℝ f x m 
goal: ((SchwartzMap.pderivCLM 𝕜 m) f) x = lineDeriv ℝ (⇑f) x m

full_name: SchwartzMap.iteratedPDeriv
code: /-- The iterated partial derivative (or directional derivative) as a continuous linear map on
Schwartz space. -/
def iteratedPDeriv {n : ℕ} : (Fin n → E) → 𝓢(E, F) →L[𝕜] 𝓢(E, F) 
goal: (Fin n → E) → SchwartzMap E F →L[𝕜] SchwartzMap E F

full_name: SchwartzMap.iteratedPDeriv_zero
code: @[simp]
theorem iteratedPDeriv_zero (m : Fin 0 → E) (f : 𝓢(E, F)) : iteratedPDeriv 𝕜 m f = f 
goal: (SchwartzMap.iteratedPDeriv 𝕜 m) f = f

full_name: SchwartzMap.iteratedPDeriv_one
code: @[simp]
theorem iteratedPDeriv_one (m : Fin 1 → E) (f : 𝓢(E, F)) :
    iteratedPDeriv 𝕜 m f = pderivCLM 𝕜 (m 0) f 
goal: (SchwartzMap.iteratedPDeriv 𝕜 m) f = (SchwartzMap.pderivCLM 𝕜 (m 0)) f

full_name: SchwartzMap.iteratedPDeriv_succ_left
code: theorem iteratedPDeriv_succ_left {n : ℕ} (m : Fin (n + 1) → E) (f : 𝓢(E, F)) :
    iteratedPDeriv 𝕜 m f = pderivCLM 𝕜 (m 0) (iteratedPDeriv 𝕜 (Fin.tail m) f) 
goal: (SchwartzMap.iteratedPDeriv 𝕜 m) f = (SchwartzMap.pderivCLM 𝕜 (m 0)) ((SchwartzMap.iteratedPDeriv 𝕜 (Fin.tail m)) f)

full_name: SchwartzMap.iteratedPDeriv_succ_right
code: theorem iteratedPDeriv_succ_right {n : ℕ} (m : Fin (n + 1) → E) (f : 𝓢(E, F)) :
    iteratedPDeriv 𝕜 m f = iteratedPDeriv 𝕜 (Fin.init m) (pderivCLM 𝕜 (m (Fin.last n)) f) 
goal: (SchwartzMap.iteratedPDeriv 𝕜 m) f =
  (SchwartzMap.iteratedPDeriv 𝕜 (Fin.init m)) ((SchwartzMap.pderivCLM 𝕜 (m (Fin.last n))) f)

full_name: SchwartzMap.iteratedPDeriv_eq_iteratedFDeriv
code: theorem iteratedPDeriv_eq_iteratedFDeriv {n : ℕ} {m : Fin n → E} {f : 𝓢(E, F)} {x : E} :
    iteratedPDeriv 𝕜 m f x = iteratedFDeriv ℝ n f x m 
goal: ((SchwartzMap.iteratedPDeriv 𝕜 m) f) x = (iteratedFDeriv ℝ n (⇑f) x) m

full_name: SchwartzMap.integrable_pow_mul_iteratedFDeriv
code: lemma integrable_pow_mul_iteratedFDeriv (f : 𝓢(D, V))
    (k n : ℕ) : Integrable (fun x ↦ ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖) μ 
goal: MeasureTheory.Integrable (fun x => ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖) μ

full_name: SchwartzMap.integrable_pow_mul
code: lemma integrable_pow_mul (f : 𝓢(D, V))
    (k : ℕ) : Integrable (fun x ↦ ‖x‖ ^ k * ‖f x‖) μ 
goal: MeasureTheory.Integrable (fun x => ‖x‖ ^ k * ‖f x‖) μ

full_name: SchwartzMap.integral_pow_mul_iteratedFDeriv_le
code: lemma integral_pow_mul_iteratedFDeriv_le (f : 𝓢(D, V)) (k n : ℕ) :
    ∫ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ∂μ ≤ 2 ^ μ.integrablePower *
      (∫ x, (1 + ‖x‖) ^ (- (μ.integrablePower : ℝ)) ∂μ) *
        (SchwartzMap.seminorm 𝕜 0 n f + SchwartzMap.seminorm 𝕜 (k + μ.integrablePower) n f) 
goal: ∫ (x : D), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ∂μ ≤
  (2 ^ μ.integrablePower * ∫ (x : D), (1 + ‖x‖) ^ (-↑μ.integrablePower) ∂μ) *
    ((SchwartzMap.seminorm 𝕜 0 n) f + (SchwartzMap.seminorm 𝕜 (k + μ.integrablePower) n) f)

full_name: SchwartzMap.integrable
code: lemma integrable (f : 𝓢(D, V)) : Integrable f μ 
goal: MeasureTheory.Integrable (⇑f) μ

full_name: SchwartzMap.integralCLM
code: /-- The integral as a continuous linear map from Schwartz space to the codomain. -/
def integralCLM : 𝓢(D, V) →L[𝕜] V 
goal: SchwartzMap D V →L[𝕜] V

full_name: SchwartzMap.integralCLM_apply
code: @[simp]
lemma integralCLM_apply (f : 𝓢(D, V)) : integralCLM 𝕜 μ f = ∫ x, f x ∂μ 
goal: (SchwartzMap.integralCLM 𝕜 μ) f = ∫ (x : D), f x ∂μ

full_name: SchwartzMap.instBoundedContinuousMapClass
code: instance instBoundedContinuousMapClass : BoundedContinuousMapClass 𝓢(E, F) E F 
goal: BoundedContinuousMapClass (SchwartzMap E F) E F

full_name: SchwartzMap.toBoundedContinuousFunction
code: /-- Schwartz functions as bounded continuous functions -/
def toBoundedContinuousFunction (f : 𝓢(E, F)) : E →ᵇ F 
goal: BoundedContinuousFunction E F

full_name: SchwartzMap.toBoundedContinuousFunction_apply
code: @[simp]
theorem toBoundedContinuousFunction_apply (f : 𝓢(E, F)) (x : E) :
    f.toBoundedContinuousFunction x = f x 
goal: f.toBoundedContinuousFunction x = f x

full_name: SchwartzMap.toContinuousMap
code: /-- Schwartz functions as continuous functions -/
def toContinuousMap (f : 𝓢(E, F)) : C(E, F) 
goal: C(E, F)

full_name: SchwartzMap.toBoundedContinuousFunctionCLM
code: /-- The inclusion map from Schwartz functions to bounded continuous functions as a continuous linear
map. -/
def toBoundedContinuousFunctionCLM : 𝓢(E, F) →L[𝕜] E →ᵇ F 
goal: SchwartzMap E F →L[𝕜] BoundedContinuousFunction E F

full_name: SchwartzMap.toBoundedContinuousFunctionCLM_apply
code: @[simp]
theorem toBoundedContinuousFunctionCLM_apply (f : 𝓢(E, F)) (x : E) :
    toBoundedContinuousFunctionCLM 𝕜 E F f x = f x 
goal: ((SchwartzMap.toBoundedContinuousFunctionCLM 𝕜 E F) f) x = f x

full_name: SchwartzMap.delta
code: /-- The Dirac delta distribution -/
def delta (x : E) : 𝓢(E, F) →L[𝕜] F 
goal: SchwartzMap E F →L[𝕜] F

full_name: SchwartzMap.delta_apply
code: @[simp]
theorem delta_apply (x₀ : E) (f : 𝓢(E, F)) : delta 𝕜 F x₀ f = f x₀ 
goal: (SchwartzMap.delta 𝕜 F x₀) f = f x₀

full_name: SchwartzMap.integralCLM_dirac_eq_delta
code: /-- Integrating against the Dirac measure is equal to the delta distribution. -/
@[simp]
theorem integralCLM_dirac_eq_delta (x : E) : integralCLM 𝕜 (dirac x) = delta 𝕜 F x 
goal: SchwartzMap.integralCLM 𝕜 (MeasureTheory.Measure.dirac x) = SchwartzMap.delta 𝕜 F x

full_name: SchwartzMap.instZeroAtInftyContinuousMapClass
code: instance instZeroAtInftyContinuousMapClass : ZeroAtInftyContinuousMapClass 𝓢(E, F) E F 
goal: ZeroAtInftyContinuousMapClass (SchwartzMap E F) E F

full_name: SchwartzMap.toZeroAtInfty
code: /-- Schwartz functions as continuous functions vanishing at infinity. -/
def toZeroAtInfty (f : 𝓢(E, F)) : C₀(E, F) 
goal: ZeroAtInftyContinuousMap E F

full_name: SchwartzMap.toZeroAtInfty_apply
code: @[simp] theorem toZeroAtInfty_apply (f : 𝓢(E, F)) (x : E) : f.toZeroAtInfty x = f x 
goal: f.toZeroAtInfty x = f x

full_name: SchwartzMap.toZeroAtInfty_toBCF
code: @[simp] theorem toZeroAtInfty_toBCF (f : 𝓢(E, F)) :
    f.toZeroAtInfty.toBCF = f.toBoundedContinuousFunction 
goal: f.toZeroAtInfty.toBCF = f.toBoundedContinuousFunction

full_name: SchwartzMap.toZeroAtInftyCLM
code: /-- The inclusion map from Schwartz functions to continuous functions vanishing at infinity as a
continuous linear map. -/
def toZeroAtInftyCLM : 𝓢(E, F) →L[𝕜] C₀(E, F) 
goal: SchwartzMap E F →L[𝕜] ZeroAtInftyContinuousMap E F

full_name: SchwartzMap.toZeroAtInftyCLM_apply
code: @[simp] theorem toZeroAtInftyCLM_apply (f : 𝓢(E, F)) (x : E) : toZeroAtInftyCLM 𝕜 E F f x = f x 
goal: ((SchwartzMap.toZeroAtInftyCLM 𝕜 E F) f) x = f x