Uncurrying continuous alternating maps

Given a continuous function f which is linear in the first argument and is alternating form in the other n arguments, this file defines a continuous alternating form ContinuousAlternatingMap.alternatizeUncurryFin f in n + 1 arguments.

This function is given by

ContinuousAlternatingMap.alternatizeUncurryFin f v =
  ∑ i : Fin (n + 1), (-1) ^ (i : ℕ) • f (v i) (removeNth i v)

Given a continuous alternating map f of n + 1 arguments, each term in the sum above written for f.curryLeft equals the original map, thus f.curryLeft.alternatizeUncurryFin = (n + 1) • f.

We do not multiply the result of alternatizeUncurryFin by (n + 1)⁻¹ so that the construction works for 𝕜-multilinear maps over any normed field 𝕜, not only a field of characteristic zero.

Main results

• ContinuousAlternatingMap.alternatizeUncurryFin_curryLeft: the round-trip formula for currying/uncurrying, see above.

• ContinuousAlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinLM_comp_of_symmetric: If f is a symmetric bilinear map taking values in the space of continuous alternating maps, then the twice uncurried f is zero.

The latter theorem will be used to prove that the second exterior derivative of a differential form is zero.

theorem ContinuousAlternatingMap.map_insertNth {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (f : E [⋀^Fin (n + 1)]→L[𝕜] F) (p : Fin (n + 1)) (x : E) (v : Fin n → E) : f (p.insertNth x v) = (-1) ^ ↑p • f (Matrix.vecCons x v)

If f is a continuous (n + 1)-multilinear alternating map, x is an element of the domain, and v is an n-vector, then the value of f at v with x inserted at the pth place equals (-1) ^ p times the value of f at v with x prepended.

theorem ContinuousAlternatingMap.neg_one_pow_smul_map_insertNth {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (f : E [⋀^Fin (n + 1)]→L[𝕜] F) (p : Fin (n + 1)) (x : E) (v : Fin n → E) : (-1) ^ ↑p • f (p.insertNth x v) = f (Matrix.vecCons x v)

theorem ContinuousAlternatingMap.neg_one_pow_smul_map_removeNth_add_eq_zero_of_eq {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (f : E [⋀^Fin n]→L[𝕜] F) {v : Fin (n + 1) → E} {i j : Fin (n + 1)} (hvij : v i = v j) (hij : i ≠ j) : (-1) ^ ↑i • f (i.removeNth v) + (-1) ^ ↑j • f (j.removeNth v) = 0

Let v be an (n + 1)-tuple with two equal elements v i = v j, i ≠ j. Let w i (resp., w j) be the vector v with ith (resp., jth) element removed. Then (-1) ^ i • f (w i) + (-1) ^ j • f (w j) = 0. This follows from the fact that these two vectors differ by a permutation of sign (-1) ^ (i + j).

These are the only two nonzero terms in the proof of map_eq_zero_of_eq in the definition of AlternatingMap.alternatizeUncurryFin.

@[irreducible]

noncomputable def ContinuousAlternatingMap.alternatizeUncurryFinCLM (𝕜 : Type u_1) (E : Type u_2) (F : Type u_3) [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} : (E →L[𝕜] E [⋀^Fin n]→L[𝕜] F) →L[𝕜] E [⋀^Fin (n + 1)]→L[𝕜] F

AlternaringMap.alternatizeUncurryFin as a continuous linear map.

Equations

• ContinuousAlternatingMap.alternatizeUncurryFinCLM 𝕜 E F = AlternatingMap.mkContinuousLinear ContinuousAlternatingMap.alternatizeUncurryFinCLM.aux✝ (↑n + 1) ⋯

theorem ContinuousAlternatingMap.norm_alternatizeUncurryFinCLM_le {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} : ‖alternatizeUncurryFinCLM 𝕜 E F‖ ≤ ↑n + 1

noncomputable def ContinuousAlternatingMap.alternatizeUncurryFin {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (f : E →L[𝕜] E [⋀^Fin n]→L[𝕜] F) : E [⋀^Fin (n + 1)]→L[𝕜] F

Given a continuous function which is linear in the first argument and is alternating in the other n arguments, build a continuous alternating form in n + 1 arguments.

The function is given by

alternatizeUncurryFin f v = ∑ i : Fin (n + 1), (-1) ^ (i : ℕ) • f (v i) (removeNth i v)

Note that the round-trip with curryLeft multiplies the form by n + 1, since we want to avoid division in this definition.

Equations

• ContinuousAlternatingMap.alternatizeUncurryFin f = (ContinuousAlternatingMap.alternatizeUncurryFinCLM 𝕜 E F) f

@[simp]

theorem ContinuousAlternatingMap.alternatizeUncurryFinCLM_apply {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (f : E →L[𝕜] E [⋀^Fin n]→L[𝕜] F) : (alternatizeUncurryFinCLM 𝕜 E F) f = alternatizeUncurryFin f

theorem ContinuousAlternatingMap.norm_alternatizeUncurryFin_le {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (f : E →L[𝕜] E [⋀^Fin n]→L[𝕜] F) : ‖alternatizeUncurryFin f‖ ≤ (↑n + 1) * ‖f‖

theorem ContinuousAlternatingMap.alternatizeUncurryFin_apply {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (f : E →L[𝕜] E [⋀^Fin n]→L[𝕜] F) (v : Fin (n + 1) → E) : (alternatizeUncurryFin f) v = ∑ i : Fin (n + 1), (-1) ^ ↑i • (f (v i)) (i.removeNth v)

theorem ContinuousAlternatingMap.toAlternatingMap_alternatizeUncurryFin {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (f : E →L[𝕜] E [⋀^Fin n]→L[𝕜] F) : (alternatizeUncurryFin f).toAlternatingMap = AlternatingMap.alternatizeUncurryFin (toAlternatingMapLinear ∘ₗ ↑f)

@[simp]

theorem ContinuousAlternatingMap.alternatizeUncurryFin_add {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (f g : E →L[𝕜] E [⋀^Fin n]→L[𝕜] F) : alternatizeUncurryFin (f + g) = alternatizeUncurryFin f + alternatizeUncurryFin g

@[simp]

theorem ContinuousAlternatingMap.alternatizeUncurryFin_curryLeft {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (f : E [⋀^Fin (n + 1)]→L[𝕜] F) : alternatizeUncurryFin f.curryLeft = (n + 1) • f

@[simp]

theorem ContinuousAlternatingMap.alternatizeUncurryFin_smul {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {S : Type u_4} [Monoid S] [DistribMulAction S F] [ContinuousConstSMul S F] [SMulCommClass 𝕜 S F] (c : S) (f : E →L[𝕜] E [⋀^Fin n]→L[𝕜] F) : alternatizeUncurryFin (c • f) = c • alternatizeUncurryFin f

theorem ContinuousAlternatingMap.alternatizeUncurryFin_constOfIsEmptyLIE_comp {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E →L[𝕜] F) : alternatizeUncurryFin ((↑{ toLinearEquiv := (constOfIsEmptyLIE 𝕜 E F (Fin 0)).toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).comp f) = (ofSubsingleton 𝕜 E F 0) f

theorem ContinuousAlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinCLM_comp_of_symmetric {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : E →L[𝕜] E →L[𝕜] E [⋀^Fin n]→L[𝕜] F} (hf : ∀ (x y : E), (f x) y = (f y) x) : alternatizeUncurryFin ((alternatizeUncurryFinCLM 𝕜 E F).comp f) = 0

If f is a symmetric continuous bilinear map taking values in the space of continuous alternating maps, then the twice uncurried f is zero.