mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : RCLike 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℂ 𝕜 E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
k n : ℕ
f : 𝓢(V, E)
x : V
p : ℕ × ℕ
a✝ : p ∈ Finset.range (n + 1) ×ˢ Finset.range (k + 1)
⊢ (p.1 + volume.integrablePower, p.2) ∈ Finset.range (n + volume.integrablePower + 1) ×ˢ Finset.range (k + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.FourierMultiplier

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : NormedSpace ℂ F
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
f : 𝓢'(E, F)
ι : Type := Fin (Module.finrank ℝ E)
b : OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E := stdOrthonormalBasis ℝ E
this : ∀ i ∈ Finset.univ, Function.HasTemperateGrowth fun x ↦ ↑(inner ℝ x (b i)) ^ 2
i : Fin (Module.finrank ℝ E)
x : 𝓢(E, ℂ)
⊢ Function.HasTemperateGrowth fun x ↦ ↑(inner ℝ x (b i))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : RCLike 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℂ 𝕜 E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
k n : ℕ
f : 𝓢(V, E)
x : V
⊢ ∑ p ∈ Finset.range (n + 1) ×ˢ Finset.range (k + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 (⇑f) v‖ ≤
    ∑ x ∈ Finset.range (n + 1) ×ˢ Finset.range (k + 1),
      (2 ^ volume.integrablePower * ∫ (x : V), (1 + ‖x‖) ^ (-↑volume.integrablePower)) * 2 *
        ((Finset.range (n + volume.integrablePower + 1) ×ˢ Finset.range (k + 1)).sup (schwartzSeminormFamily 𝕜 V E)) f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : RCLike 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℂ 𝕜 E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
k n : ℕ
f : 𝓢(V, E)
x : V
⊢ ∑ x ∈ Finset.range (n + 1) ×ˢ Finset.range (k + 1),
      (2 ^ volume.integrablePower * ∫ (x : V), (1 + ‖x‖) ^ (-↑volume.integrablePower)) * 2 *
        ((Finset.range (n + volume.integrablePower + 1) ×ˢ Finset.range (k + 1)).sup (schwartzSeminormFamily 𝕜 V E)) f =
    ↑(Finset.range (n + 1) ×ˢ Finset.range (k + 1)).card *
      (2 ^ volume.integrablePower *
        ((∫ (x : V), (1 + ‖x‖) ^ (-↑volume.integrablePower)) *
          (2 *
            ((Finset.range (n + volume.integrablePower + 1) ×ˢ Finset.range (k + 1)).sup (schwartzSeminormFamily 𝕜 V E))
              f)))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : RCLike 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℂ 𝕜 E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
⊢ 𝓢(V, E) →L[𝕜] 𝓢(V, E)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : 𝓢(V, E)
⊢ ⇑(𝓕⁻ f) = 𝓕⁻ ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : 𝓢(V, E)
⊢ 𝓕⁻ f = (compCLMOfContinuousLinearEquiv ℂ ↑(LinearIsometryEquiv.neg ℝ)) (𝓕 f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.FourierMultiplier

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : NormedSpace ℂ F
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
f : 𝓢'(E, F)
⊢ Δ f = -(2 * π) ^ 2 • (fourierMultiplierCLM F fun x ↦ ↑(‖x‖ ^ 2)) f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : RCLike 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : SMulCommClass ℂ 𝕜 E
V : Type u_3
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : CompleteSpace E
⊢ ∀ (f : 𝓢(V, E)), 𝓕⁻ (𝓕 f) = f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : RCLike 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : SMulCommClass ℂ 𝕜 E
V : Type u_3
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : CompleteSpace E
⊢ ∀ (f : 𝓢(V, E)), 𝓕 (𝓕⁻ f) = f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: V : Type u_3
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : InnerProductSpace ℝ V
inst✝⁹ : FiniteDimensional ℝ V
inst✝⁸ : MeasurableSpace V
inst✝⁷ : BorelSpace V
𝕜' : Type u_4
inst✝⁶ : NormedField 𝕜'
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G : Type u_6
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℂ G
inst✝¹ : NormedSpace 𝕜' G
inst✝ : SMulCommClass ℝ 𝕜' G
f : 𝓢(V, F →L[ℝ] G)
m : F
⊢ 𝓕 ((SchwartzMap.evalCLM 𝕜' V G m) f) = (SchwartzMap.evalCLM 𝕜' V G m) (𝓕 f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : RCLike 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℂ 𝕜 E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : 𝓢(V, E)
⊢ (fderivCLM 𝕜 V E) (𝓕 f) = 𝓕 (-(2 * ↑π * Complex.I) • (smulRightCLM ℂ E (innerSL ℝ)) f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : RCLike 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℂ 𝕜 E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : 𝓢(V, E)
⊢ 𝓕 ((fderivCLM 𝕜 V E) f) = (2 * ↑π * Complex.I) • (smulRightCLM ℂ E (innerSL ℝ)) (𝓕 f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : 𝓢(V, E)
m : V
⊢ ∂_{m} (𝓕 f) = 𝓕 (-(2 * ↑π * Complex.I) • (smulLeftCLM E fun x ↦ inner ℝ x m) f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : 𝓢(V, E)
m : V
⊢ 𝓕 (∂_{m} f) = (2 * ↑π * Complex.I) • (smulLeftCLM E fun x ↦ inner ℝ x m) (𝓕 f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : 𝓢(V, E)
m : V
⊢ ∂_{m} (𝓕⁻ f) = 𝓕⁻ ((2 * ↑π * Complex.I) • (smulLeftCLM E fun x ↦ inner ℝ x m) f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : 𝓢(V, E)
m : V
⊢ Function.HasTemperateGrowth fun x ↦ inner ℝ x m

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : 𝓢(V, E)
m : V
⊢ 𝓕⁻ (∂_{m} f) = -(2 * ↑π * Complex.I) • (smulLeftCLM E fun x ↦ inner ℝ x m) (𝓕⁻ f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℂ E
V : Type u_3
inst✝¹⁰ : NormedAddCommGroup V
inst✝⁹ : InnerProductSpace ℝ V
inst✝⁸ : FiniteDimensional ℝ V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℂ G
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
f : 𝓢(V, E)
g : 𝓢(V, F)
M : E →L[ℂ] F →L[ℂ] G
⊢ ∫ (ξ : V), (M ((𝓕 f) ξ)) (g ξ) = ∫ (x : V), (M (f x)) ((𝓕 g) x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℂ E
V : Type u_3
inst✝¹⁰ : NormedAddCommGroup V
inst✝⁹ : InnerProductSpace ℝ V
inst✝⁸ : FiniteDimensional ℝ V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℂ G
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
f : 𝓢(V, E)
g : 𝓢(V, F)
M : E →L[ℂ] F →L[ℂ] G
⊢ ∫ (ξ : V), (M ((𝓕⁻ f) ξ)) (g ξ) = ∫ (x : V), (M (f x)) ((𝓕⁻ g) x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℂ E
V : Type u_3
inst✝¹⁰ : NormedAddCommGroup V
inst✝⁹ : InnerProductSpace ℝ V
inst✝⁸ : FiniteDimensional ℝ V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℂ G
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
f : 𝓢(V, E)
g : 𝓢(V, F)
M : E →L⋆[ℂ] F →L[ℂ] G
⊢ ∫ (ξ : V), (M ((𝓕 f) ξ)) (g ξ) = ∫ (x : V), (M (f x)) ((𝓕⁻ g) x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℂ E
V : Type u_3
inst✝¹⁰ : NormedAddCommGroup V
inst✝⁹ : InnerProductSpace ℝ V
inst✝⁸ : FiniteDimensional ℝ V
inst✝⁷ : MeasurableSpace V
inst✝⁶ : BorelSpace V
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℂ G
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
f : 𝓢(V, E)
g : 𝓢(V, F)
M : E →L⋆[ℂ] F →L[ℂ] G
⊢ ∫ (ξ : V), (M ((𝓕 f) ξ)) ((𝓕 g) ξ) = ∫ (x : V), (M (f x)) (g x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace ℝ V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : 𝓢(V, F)
x : V
⊢ ‖(𝓕 f) x‖ = ‖∫ (v : V), 𝐞 (-inner ℝ v x) • f v‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace ℝ V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : 𝓢(V, F)
x : V
⊢ ∫ (v : V), ‖𝐞 (-inner ℝ v x) • f v‖ = ‖f.toLp 1 volume‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace ℝ V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : 𝓢(V, F)
⊢ 0 ≤ ‖f.toLp 1 volume‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace ℝ V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : 𝓢(V, F)
⊢ ‖(𝓕 f).toBoundedContinuousFunction‖ ≤ ‖f.toLp 1 volume‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace ℝ V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : 𝓢(V, F)
⊢ ∀ (x : V), ‖x‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑(𝓕 f)) x‖ ≤ ‖f.toLp 1 volume‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : FiniteDimensional ℝ V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
H : Type u_4
inst✝² : NormedAddCommGroup H
inst✝¹ : InnerProductSpace ℂ H
inst✝ : CompleteSpace H
f : 𝓢(V, H)
⊢ ∫ (ξ : V), ‖(𝓕 f) ξ‖ ^ 2 = ∫ (x : V), ‖f x‖ ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : FiniteDimensional ℝ V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
H : Type u_4
inst✝² : NormedAddCommGroup H
inst✝¹ : InnerProductSpace ℂ H
inst✝ : CompleteSpace H
f g : 𝓢(V, H)
⊢ ⟪(𝓕 f).toLp 2 volume, (𝓕 g).toLp 2 volume⟫ = ⟪f.toLp 2 volume, g.toLp 2 volume⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Goal state before:
pp: V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : FiniteDimensional ℝ V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
H : Type u_4
inst✝² : NormedAddCommGroup H
inst✝¹ : InnerProductSpace ℂ H
inst✝ : CompleteSpace H
f : 𝓢(V, H)
⊢ ‖(𝓕 f).toLp 2 volume‖ = ‖f.toLp 2 volume‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : TopologicalSpace α
inst✝¹ : Zero β
f : F → V
s : Set α
inst✝ : Zero V
⊢ ¬IsVanishingOn f s ↔ ∃ u, tsupport ⇑u ⊆ s ∧ f u ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : TopologicalSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
x : α
⊢ x ∈ dsupport f ↔ ∀ (s : Set α), IsVanishingOn f sᶜ → IsClosed[inst✝²] s → x ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : TopologicalSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
⊢ (dsupport f)ᶜ = ⋃₀ {a | IsVanishingOn f a ∧ IsOpen[inst✝²] a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : TopologicalSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
x : α
⊢ x ∉ dsupport f ↔ ∃ s, IsVanishingOn f s ∧ IsOpen[inst✝²] s ∧ x ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : TopologicalSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
x : α
⊢ x ∈ dsupport f ↔ ∀ (s : Set α), x ∈ s → IsOpen[inst✝²] s → ¬IsVanishingOn f s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : TopologicalSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
x : α
⊢ x ∈ dsupport f ↔ ∀ (s : Set α), x ∈ s → IsOpen[inst✝²] s → ∃ u, tsupport ⇑u ⊆ s ∧ f u ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : TopologicalSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
x : α
⊢ x ∈ dsupport f ↔ ∃ᶠ (u : Set α) in (𝓝 x).smallSets, ¬IsVanishingOn f u

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : TopologicalSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
ι : Sort u_11
p : ι → Prop
s : ι → Set α
x : α
hl : (𝓝 x).HasBasis p s
⊢ x ∈ dsupport f ↔ ∀ (i : ι), p i → ¬IsVanishingOn f (s i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : TopologicalSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
x : α
⊢ x ∉ dsupport f ↔ ∀ᶠ (u : Set α) in (𝓝 x).smallSets, IsVanishingOn f u

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : TopologicalSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
ι : Sort u_11
p : ι → Prop
s : ι → Set α
x : α
hl : (𝓝 x).HasBasis p s
⊢ x ∉ dsupport f ↔ ∃ i, p i ∧ IsVanishingOn f (s i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : TopologicalSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
⊢ IsClosed[inst✝²] (dsupport f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : TopologicalSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
s : Set α
h : IsVanishingOn f s
s_open : IsOpen[inst✝²] s
⊢ Disjoint s (dsupport f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: α : Type u_2
β : Type u_3
F : Type u_6
V : Type u_10
inst✝³ : FunLike F α β
inst✝² : PseudoMetricSpace α
inst✝¹ : Zero β
inst✝ : Zero V
f : F → V
⊢ (dsupport f)ᶜ =
    ⋃₀ {a | IsVanishingOn f a ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a ∧ Bornology.IsBounded a}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ F
f : 𝓢'(E, F)
s : Set E
hf : IsVanishingOn (⇑f) s
g : E → ℂ
hg : Function.HasTemperateGrowth g
⊢ IsVanishingOn (⇑((TemperedDistribution.smulLeftCLM F g) f)) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ F
f : 𝓢'(E, F)
s : Set E
hf : IsVanishingOn (⇑f) s
m : E
⊢ IsVanishingOn (⇑(∂_{m} f)) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ F
f : 𝓢'(E, F)
s : Set E
n : ℕ
hf : IsVanishingOn (⇑f) s
m : Fin n → E
⊢ IsVanishingOn (⇑(∂^{m} f)) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
⊢ IsVanishingOn ⇑(TemperedDistribution.delta x) {x}ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ F
f : 𝓢'(E, F)
g : E → ℂ
hg : Function.HasTemperateGrowth g
⊢ dsupport ⇑((TemperedDistribution.smulLeftCLM F g) f) ⊆ dsupport ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ F
f : 𝓢'(E, F)
m : E
⊢ dsupport ⇑(∂_{m} f) ⊆ dsupport ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ F
f : 𝓢'(E, F)
n : ℕ
m : Fin n → E
⊢ dsupport ⇑(∂^{m} f) ⊆ dsupport ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
s : Set E
hx : x ∈ s
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
u : E → ℝ
h₁ : tsupport u ⊆ s
h₂ : HasCompactSupport u
h₃ : ContDiff ℝ (↑⊤) u
h₄ : u x = 1
h₁' : tsupport (⇑Complex.ofRealCLM ∘ u) ⊆ s
h₂' : HasCompactSupport (⇑Complex.ofRealCLM ∘ u)
⊢ (TemperedDistribution.delta x) (h₂'.toSchwartzMap ⋯) ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ dsupport ⇑(TemperedDistribution.delta x) = {x}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_11
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
Ω : Opens E
F : Type u_12
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
f : 𝓓'(Ω, F)
s : Set E
hf : IsVanishingOn (⇑f) s
m : E
⊢ IsVanishingOn (⇑(∂_{m} f)) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_11
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
Ω : Opens E
F : Type u_12
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
f : 𝓓'(Ω, F)
s : Set E
n : ℕ
hf : IsVanishingOn (⇑f) s
m : Fin n → E
⊢ IsVanishingOn (⇑(∂^{m} f)) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : NormedSpace ℂ F
⊢ besselPotential E F 0 = ContinuousLinearMap.id ℂ 𝓢'(E, F)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.Support

Goal state before:
pp: E : Type u_11
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
Ω : Opens E
n : ℕ∞
x : E
⊢ IsVanishingOn ⇑(delta x) {x}ᶜ

Tactic:
by

Goal state after: