Skip to content

chore(Analysis/Distribution): split off derivatives from SchwartzSpace and create new folder #34027

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Open
mcdoll wants to merge 5 commits into
base: master
Choose a base branch
from
Open

chore(Analysis/Distribution): split off derivatives from SchwartzSpace and create new folder #34027

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
5 commits
Select commit Hold shift + click to select a range
e22fede
moving
mcdoll Jan 16, 2026
e50c783
name
mcdoll Jan 16, 2026
a719f64
deprecations
mcdoll Jan 16, 2026
3601982
Revert "deprecations"
mcdoll Jan 16, 2026
f75143a
Merge branch 'master' into split_schwartz
mcdoll Jan 16, 2026
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
5 changes: 3 additions & 2 deletions Mathlib.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -1787,8 +1787,9 @@ public import Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff
public import Mathlib.Analysis.Distribution.ContDiffMapSupportedIn
public import Mathlib.Analysis.Distribution.DerivNotation
public import Mathlib.Analysis.Distribution.Distribution
public import Mathlib.Analysis.Distribution.FourierSchwartz
public import Mathlib.Analysis.Distribution.SchwartzSpace
public import Mathlib.Analysis.Distribution.SchwartzSpace.Basic
public import Mathlib.Analysis.Distribution.SchwartzSpace.Deriv
public import Mathlib.Analysis.Distribution.SchwartzSpace.Fourier
public import Mathlib.Analysis.Distribution.TemperateGrowth
public import Mathlib.Analysis.Distribution.TemperedDistribution
public import Mathlib.Analysis.Distribution.TestFunction
Expand Down
224 changes: 2 additions & 222 deletions .../Analysis/Distribution/SchwartzSpace.lean → ...sis/Distribution/SchwartzSpace/Basic.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -5,16 +5,12 @@ Authors: Moritz Doll
-/
module

public import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
public import Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts
public import Mathlib.Analysis.LocallyConvex.WithSeminorms
public import Mathlib.Analysis.Normed.Group.ZeroAtInfty
public import Mathlib.Analysis.Normed.Lp.SmoothApprox
public import Mathlib.Analysis.SpecialFunctions.Pow.Real
public import Mathlib.Analysis.Distribution.DerivNotation
public import Mathlib.Analysis.Distribution.TemperateGrowth
public import Mathlib.Topology.Algebra.UniformFilterBasis
public import Mathlib.MeasureTheory.Integral.IntegralEqImproper
public import Mathlib.MeasureTheory.Function.L2Space

/-!
Expand All @@ -25,7 +21,7 @@ functions $f : ℝ^n → ℂ$ such that there exists $C_{αβ} > 0$ with $$|x^α
all $x ∈ ℝ^n$ and for all multiindices $α, β$.
In mathlib, we use a slightly different approach and define the Schwartz space as all
smooth functions `f : E → F`, where `E` and `F` are real normed vector spaces such that for all
natural numbers `k` and `n` we have uniform bounds `‖x‖^k * ‖iteratedFDeriv ℝ n f x‖ < C`.
natural numbers `k` and `n` we have uniform bounds `‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ < C`.
This approach completely avoids using partial derivatives as well as polynomials.
We construct the topology on the Schwartz space by a family of seminorms, which are the best
constants in the above estimates. The abstract theory of topological vector spaces developed in
Expand All @@ -40,10 +36,6 @@ Schwartz space into a locally convex topological vector space.
* `SchwartzMap.compCLM`: Composition with a function on the right as a continuous linear map
`𝓢(E, F) →L[𝕜] 𝓢(D, F)`, provided that the function is temperate and grows polynomially near
infinity
* `SchwartzMap.fderivCLM`: The differential as a continuous linear map
`𝓢(E, F) →L[𝕜] 𝓢(E, E →L[ℝ] F)`
* `SchwartzMap.derivCLM`: The one-dimensional derivative as a continuous linear map
`𝓢(ℝ, F) →L[𝕜] 𝓢(ℝ, F)`
* `SchwartzMap.integralCLM`: Integration as a continuous linear map `𝓢(ℝ, F) →L[ℝ] F`

## Main statements
Expand All @@ -66,9 +58,7 @@ The implementation of the seminorms is taken almost literally from `ContinuousLi
Schwartz space, tempered distributions
-/

@[expose] public section

noncomputable section
@[expose] public noncomputable section

open scoped Nat NNReal ContDiff

Expand Down Expand Up @@ -1002,121 +992,6 @@ theorem smulLeftCLM_compCLMOfContinuousLinearEquiv {u : D → 𝕜'} (hu : u.Has

end Comp

section Derivatives

/-! ### Derivatives of Schwartz functions -/

variable (𝕜)
variable [RCLike 𝕜] [NormedSpace 𝕜 F]

variable (F) in
/-- The 1-dimensional derivative on Schwartz space as a continuous `𝕜`-linear map. -/
def derivCLM : 𝓢(ℝ, F) →L[𝕜] 𝓢(ℝ, F) :=
mkCLM (deriv ·) (fun f g _ => deriv_add f.differentiableAt g.differentiableAt)
(fun a f _ => deriv_const_smul a f.differentiableAt)
(fun f => (contDiff_succ_iff_deriv.mp (f.smooth ⊤)).2.2) fun ⟨k, n⟩ =>
⟨{⟨k, n + 1⟩}, 1, zero_le_one, fun f x => by
simpa only [Real.norm_eq_abs, Finset.sup_singleton, schwartzSeminormFamily_apply, one_mul,
norm_iteratedFDeriv_eq_norm_iteratedDeriv, ← iteratedDeriv_succ'] using
f.le_seminorm' 𝕜 k (n + 1) x⟩

@[simp]
theorem derivCLM_apply (f : 𝓢(ℝ, F)) (x : ℝ) : derivCLM 𝕜 F f x = deriv f x :=
rfl

theorem hasDerivAt (f : 𝓢(ℝ, F)) (x : ℝ) : HasDerivAt f (deriv f x) x :=
f.differentiableAt.hasDerivAt

variable [SMulCommClass ℝ 𝕜 F]

open LineDeriv

variable (E F) in
/-- 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) :=
mkCLM (fderiv ℝ ·) (fun f g _ => fderiv_add f.differentiableAt g.differentiableAt)
(fun a f _ => fderiv_const_smul f.differentiableAt a)
(fun f => (contDiff_succ_iff_fderiv.mp (f.smooth ⊤)).2.2) fun ⟨k, n⟩ =>
⟨{⟨k, n + 1⟩}, 1, zero_le_one, fun f x => by
simpa only [schwartzSeminormFamily_apply, Seminorm.comp_apply, Finset.sup_singleton,
one_smul, norm_iteratedFDeriv_fderiv, one_mul] using f.le_seminorm 𝕜 k (n + 1) x⟩

@[simp]
theorem fderivCLM_apply (f : 𝓢(E, F)) (x : E) : fderivCLM 𝕜 E F f x = fderiv ℝ f x :=
rfl

theorem hasFDerivAt (f : 𝓢(E, F)) (x : E) : HasFDerivAt f (fderiv ℝ f x) x :=
f.differentiableAt.hasFDerivAt

/-- The partial derivative (or directional derivative) in the direction `m : E` as a
continuous linear map on Schwartz space. -/
instance instLineDeriv : LineDeriv E 𝓢(E, F) 𝓢(E, F) where
lineDerivOp m f := (SchwartzMap.evalCLM ℝ E F m ∘L fderivCLM ℝ E F) f

instance instLineDerivAdd : LineDerivAdd E 𝓢(E, F) 𝓢(E, F) where
lineDerivOp_add m := (SchwartzMap.evalCLM ℝ E F m ∘L fderivCLM ℝ E F).map_add

instance instLineDerivSMul : LineDerivSMul 𝕜 E 𝓢(E, F) 𝓢(E, F) where
lineDerivOp_smul m := (SchwartzMap.evalCLM 𝕜 E F m ∘L fderivCLM 𝕜 E F).map_smul

instance instContinuousLineDeriv : ContinuousLineDeriv E 𝓢(E, F) 𝓢(E, F) where
continuous_lineDerivOp m := (SchwartzMap.evalCLM ℝ E F m ∘L fderivCLM ℝ E F).continuous

open LineDeriv

theorem lineDerivOpCLM_eq (m : E) :
lineDerivOpCLM 𝕜 𝓢(E, F) m = SchwartzMap.evalCLM 𝕜 E F m ∘L fderivCLM 𝕜 E F := rfl

@[deprecated (since := "2025-11-25")]
alias pderivCLM := lineDerivOpCLM

@[deprecated (since := "2025-11-25")]
alias pderivCLM_apply := LineDeriv.lineDerivOpCLM_apply

theorem lineDerivOp_apply (m : E) (f : 𝓢(E, F)) (x : E) : ∂_{m} f x = lineDeriv ℝ f x m :=
f.differentiableAt.lineDeriv_eq_fderiv.symm

theorem lineDerivOp_apply_eq_fderiv (m : E) (f : 𝓢(E, F)) (x : E) :
∂_{m} f x = fderiv ℝ f x m := rfl

variable [NormedAddCommGroup D] [NormedSpace ℝ D]

theorem lineDerivOp_compCLMOfContinuousLinearEquiv (m : D) (g : D ≃L[ℝ] E) (f : 𝓢(E, F)) :
∂_{m} (compCLMOfContinuousLinearEquiv 𝕜 g f) =
compCLMOfContinuousLinearEquiv 𝕜 g (∂_{g m} f) := by
ext x
simp [lineDerivOp_apply_eq_fderiv, ContinuousLinearEquiv.comp_right_fderiv]

@[deprecated (since := "2025-11-25")]
alias iteratedPDeriv := LineDeriv.iteratedLineDerivOpCLM

@[deprecated (since := "2025-11-25")]
alias iteratedPDeriv_zero := LineDeriv.iteratedLineDerivOp_zero

@[deprecated (since := "2025-11-25")]
alias iteratedPDeriv_one := LineDeriv.iteratedLineDerivOp_one

@[deprecated (since := "2025-11-25")]
alias iteratedPDeriv_succ_left := LineDeriv.iteratedLineDerivOp_succ_left

@[deprecated (since := "2025-11-25")]
alias iteratedPDeriv_succ_right := LineDeriv.iteratedLineDerivOp_succ_right

theorem iteratedLineDerivOp_eq_iteratedFDeriv {n : ℕ} {m : Fin n → E} {f : 𝓢(E, F)} {x : E} :
∂^{m} f x = iteratedFDeriv ℝ n f x m := by
induction n generalizing x with
| zero => simp
| succ n ih =>
rw [iteratedLineDerivOp_succ_left, iteratedFDeriv_succ_apply_left,
← fderiv_continuousMultilinear_apply_const_apply]
· simp only [lineDerivOp_apply_eq_fderiv, ← ih]
· exact (f.smooth ⊤).differentiable_iteratedFDeriv (mod_cast ENat.coe_lt_top n) x

@[deprecated (since := "2025-11-25")]
alias iteratedPDeriv_eq_iteratedFDeriv := iteratedLineDerivOp_eq_iteratedFDeriv

end Derivatives

section Integration

/-! ### Integration -/
Expand Down Expand Up @@ -1424,99 +1299,4 @@ theorem inner_toL2_toL2_eq (f g : 𝓢(H, V)) (μ : Measure H := by volume_tac)

end L2

section integration_by_parts

open ENNReal MeasureTheory

section one_dim

variable [NormedAddCommGroup V] [NormedSpace ℝ V]

/-- Integration by parts of Schwartz functions for the 1-dimensional derivative.

Version for a general bilinear map. -/
theorem integral_bilinear_deriv_right_eq_neg_left (f : 𝓢(ℝ, E)) (g : 𝓢(ℝ, F))
(L : E →L[ℝ] F →L[ℝ] V) :
∫ (x : ℝ), L (f x) (deriv g x) = -∫ (x : ℝ), L (deriv f x) (g x) :=
MeasureTheory.integral_bilinear_hasDerivAt_right_eq_neg_left_of_integrable
f.hasDerivAt g.hasDerivAt (pairing L f (derivCLM ℝ F g)).integrable
(pairing L (derivCLM ℝ E f) g).integrable (pairing L f g).integrable

variable [NormedRing 𝕜] [NormedSpace ℝ 𝕜] [IsScalarTower ℝ 𝕜 𝕜] [SMulCommClass ℝ 𝕜 𝕜] in
/-- Integration by parts of Schwartz functions for the 1-dimensional derivative.

Version for multiplication of scalar-valued Schwartz functions. -/
theorem integral_mul_deriv_eq_neg_deriv_mul (f : 𝓢(ℝ, 𝕜)) (g : 𝓢(ℝ, 𝕜)) :
∫ (x : ℝ), f x * (deriv g x) = -∫ (x : ℝ), deriv f x * (g x) :=
integral_bilinear_deriv_right_eq_neg_left f g (ContinuousLinearMap.mul ℝ 𝕜)

variable [RCLike 𝕜] [NormedSpace 𝕜 F] [NormedSpace 𝕜 V]

/-- Integration by parts of Schwartz functions for the 1-dimensional derivative.

Version for a Schwartz function with values in continuous linear maps. -/
theorem integral_smul_deriv_right_eq_neg_left (f : 𝓢(ℝ, 𝕜)) (g : 𝓢(ℝ, F)) :
∫ (x : ℝ), f x • deriv g x = -∫ (x : ℝ), deriv f x • g x :=
integral_bilinear_deriv_right_eq_neg_left f g (ContinuousLinearMap.lsmul ℝ 𝕜)

/-- Integration by parts of Schwartz functions for the 1-dimensional derivative.

Version for a Schwartz function with values in continuous linear maps. -/
theorem integral_clm_comp_deriv_right_eq_neg_left (f : 𝓢(ℝ, F →L[𝕜] V)) (g : 𝓢(ℝ, F)) :
∫ (x : ℝ), f x (deriv g x) = -∫ (x : ℝ), deriv f x (g x) :=
integral_bilinear_deriv_right_eq_neg_left f g
((ContinuousLinearMap.id 𝕜 (F →L[𝕜] V)).bilinearRestrictScalars ℝ)

end one_dim

variable [NormedAddCommGroup V] [NormedSpace ℝ V]
[NormedAddCommGroup D] [NormedSpace ℝ D]
[MeasurableSpace D] {μ : Measure D} [BorelSpace D] [FiniteDimensional ℝ D] [μ.IsAddHaarMeasure]

open scoped LineDeriv

/-- Integration by parts of Schwartz functions for directional derivatives.

Version for a general bilinear map. -/
theorem integral_bilinear_lineDerivOp_right_eq_neg_left (f : 𝓢(D, E)) (g : 𝓢(D, F))
(L : E →L[ℝ] F →L[ℝ] V) (v : D) :
∫ (x : D), L (f x) (∂_{v} g x) ∂μ = -∫ (x : D), L (∂_{v} f x) (g x) ∂μ := by
apply integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable (v := v)
(bilinLeftCLM L g.hasTemperateGrowth _).integrable
(bilinLeftCLM L (∂_{v} g).hasTemperateGrowth _).integrable
(bilinLeftCLM L g.hasTemperateGrowth _).integrable
all_goals
intro x
exact (hasFDerivAt _ x).hasLineDerivAt v

variable [NormedRing 𝕜] [NormedSpace ℝ 𝕜] [IsScalarTower ℝ 𝕜 𝕜] [SMulCommClass ℝ 𝕜 𝕜] in
/-- Integration by parts of Schwartz functions for directional derivatives.

Version for multiplication of scalar-valued Schwartz functions. -/
theorem integral_mul_lineDerivOp_right_eq_neg_left (f : 𝓢(D, 𝕜)) (g : 𝓢(D, 𝕜)) (v : D) :
∫ (x : D), f x * ∂_{v} g x ∂μ = -∫ (x : D), ∂_{v} f x * g x ∂μ :=
integral_bilinear_lineDerivOp_right_eq_neg_left f g (ContinuousLinearMap.mul ℝ 𝕜) v

variable [RCLike 𝕜] [NormedSpace 𝕜 F] [NormedSpace 𝕜 V]

/-- Integration by parts of Schwartz functions for directional derivatives.

Version for scalar multiplication. -/
theorem integral_smul_lineDerivOp_right_eq_neg_left (f : 𝓢(D, 𝕜)) (g : 𝓢(D, F)) (v : D) :
∫ (x : D), f x • ∂_{v} g x ∂μ = -∫ (x : D), ∂_{v} f x • g x ∂μ :=
integral_bilinear_lineDerivOp_right_eq_neg_left f g (ContinuousLinearMap.lsmul ℝ 𝕜) v

/-- Integration by parts of Schwartz functions for directional derivatives.

Version for a Schwartz function with values in continuous linear maps. -/
theorem integral_clm_comp_lineDerivOp_right_eq_neg_left (f : 𝓢(D, F →L[𝕜] V)) (g : 𝓢(D, F))
(v : D) : ∫ (x : D), f x (∂_{v} g x) ∂μ = -∫ (x : D), ∂_{v} f x (g x) ∂μ :=
integral_bilinear_lineDerivOp_right_eq_neg_left f g
((ContinuousLinearMap.id 𝕜 (F →L[𝕜] V)).bilinearRestrictScalars ℝ) v

end integration_by_parts


end SchwartzMap

set_option linter.style.longFile 1700
Loading