Continuous multilinear maps

We define continuous multilinear maps as maps from (i : ι) → M₁ i to M₂ which are multilinear and continuous, by extending the space of multilinear maps with a continuity assumption. Here, M₁ i and M₂ are modules over a ring R, and ι is an arbitrary type, and all these spaces are also topological spaces.

Main definitions

• ContinuousMultilinearMap R M₁ M₂ is the space of continuous multilinear maps from (i : ι) → M₁ i to M₂. We show that it is an R-module.

Implementation notes

We mostly follow the API of multilinear maps.

Notation

We introduce the notation M [×n]→L[R] M' for the space of continuous n-multilinear maps from M^n to M'. This is a particular case of the general notion (where we allow varying dependent types as the arguments of our continuous multilinear maps), but arguably the most important one, especially when defining iterated derivatives.

structure ContinuousMultilinearMap (R : Type u) {ι : Type v} (M₁ : ι → Type w₁) (M₂ : Type w₂) [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] extends MultilinearMap R M₁ M₂ : Type (max (max v w₁) w₂)

Continuous multilinear maps over the ring R, from ∀ i, M₁ i to M₂ where M₁ i and M₂ are modules over R with a topological structure. In applications, there will be compatibility conditions between the algebraic and the topological structures, but this is not needed for the definition.

• toFun : ((i : ι) → M₁ i) → M₂
• map_update_add' [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i) : self.toFun (Function.update m i (x + y)) = self.toFun (Function.update m i x) + self.toFun (Function.update m i y)
• map_update_smul' [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i) : self.toFun (Function.update m i (c • x)) = c • self.toFun (Function.update m i x)
• cont : Continuous self.toFun

  Continuous multilinear maps over the ring R, from ∀ i, M₁ i to M₂ where M₁ i and M₂ are modules over R with a topological structure. In applications, there will be compatibility conditions between the algebraic and the topological structures, but this is not needed for the definition.

def «term_[×_]→L[_]_» : Lean.TrailingParserDescr

Continuous multilinear maps over the ring R, from ∀ i, M₁ i to M₂ where M₁ i and M₂ are modules over R with a topological structure. In applications, there will be compatibility conditions between the algebraic and the topological structures, but this is not needed for the definition.

Equations

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

theorem ContinuousMultilinearMap.toMultilinearMap_injective {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] : Function.Injective ContinuousMultilinearMap.toMultilinearMap

instance ContinuousMultilinearMap.funLike {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] : FunLike (ContinuousMultilinearMap R M₁ M₂) ((i : ι) → M₁ i) M₂

Equations

• ContinuousMultilinearMap.funLike = { coe := fun (f : ContinuousMultilinearMap R M₁ M₂) => f.toFun, coe_injective' := ⋯ }

instance ContinuousMultilinearMap.continuousMapClass {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] : ContinuousMapClass (ContinuousMultilinearMap R M₁ M₂) ((i : ι) → M₁ i) M₂

def ContinuousMultilinearMap.Simps.apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (L₁ : ContinuousMultilinearMap R M₁ M₂) (v : (i : ι) → M₁ i) : M₂

See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections.

Equations

• ContinuousMultilinearMap.Simps.apply L₁ v = L₁ v

theorem ContinuousMultilinearMap.coe_continuous {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) : Continuous ⇑f

@[simp]

theorem ContinuousMultilinearMap.coe_coe {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) : ⇑f.toMultilinearMap = ⇑f

theorem ContinuousMultilinearMap.ext {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {f f' : ContinuousMultilinearMap R M₁ M₂} (H : ∀ (x : (i : ι) → M₁ i), f x = f' x) : f = f'

@[simp]

theorem ContinuousMultilinearMap.map_update_add {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i) : f (Function.update m i (x + y)) = f (Function.update m i x) + f (Function.update m i y)

@[deprecated ContinuousMultilinearMap.map_update_add]

theorem ContinuousMultilinearMap.map_add {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i) : f (Function.update m i (x + y)) = f (Function.update m i x) + f (Function.update m i y)

Alias of ContinuousMultilinearMap.map_update_add.

@[simp]

theorem ContinuousMultilinearMap.map_update_smul {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i) : f (Function.update m i (c • x)) = c • f (Function.update m i x)

@[deprecated ContinuousMultilinearMap.map_update_smul]

theorem ContinuousMultilinearMap.map_smul {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i) : f (Function.update m i (c • x)) = c • f (Function.update m i x)

Alias of ContinuousMultilinearMap.map_update_smul.

theorem ContinuousMultilinearMap.map_coord_zero {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) {m : (i : ι) → M₁ i} (i : ι) (h : m i = 0) : f m = 0

@[simp]

theorem ContinuousMultilinearMap.map_zero {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [Nonempty ι] : f 0 = 0

instance ContinuousMultilinearMap.instZero {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] : Zero (ContinuousMultilinearMap R M₁ M₂)

Equations

• ContinuousMultilinearMap.instZero = { zero := let __src := 0; { toMultilinearMap := __src, cont := ⋯ } }

instance ContinuousMultilinearMap.instInhabited {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] : Inhabited (ContinuousMultilinearMap R M₁ M₂)

Equations

• ContinuousMultilinearMap.instInhabited = { default := 0 }

@[simp]

theorem ContinuousMultilinearMap.zero_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (m : (i : ι) → M₁ i) : 0 m = 0

@[simp]

theorem ContinuousMultilinearMap.toMultilinearMap_zero {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] : ContinuousMultilinearMap.toMultilinearMap 0 = 0

instance ContinuousMultilinearMap.instSMul {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {R' : Type u_1} {A : Type u_3} [Monoid R'] [Semiring A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [DistribMulAction R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] : SMul R' (ContinuousMultilinearMap A M₁ M₂)

Equations

• ContinuousMultilinearMap.instSMul = { smul := fun (c : R') (f : ContinuousMultilinearMap A M₁ M₂) => let __src := c • f.toMultilinearMap; { toMultilinearMap := __src, cont := ⋯ } }

@[simp]

theorem ContinuousMultilinearMap.smul_apply {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {R' : Type u_1} {A : Type u_3} [Monoid R'] [Semiring A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [DistribMulAction R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] (f : ContinuousMultilinearMap A M₁ M₂) (c : R') (m : (i : ι) → M₁ i) : (c • f) m = c • f m

@[simp]

theorem ContinuousMultilinearMap.toMultilinearMap_smul {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {R' : Type u_1} {A : Type u_3} [Monoid R'] [Semiring A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [DistribMulAction R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] (c : R') (f : ContinuousMultilinearMap A M₁ M₂) : (c • f).toMultilinearMap = c • f.toMultilinearMap

instance ContinuousMultilinearMap.instSMulCommClass {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {R' : Type u_1} {R'' : Type u_2} {A : Type u_3} [Monoid R'] [Monoid R''] [Semiring A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [DistribMulAction R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] [DistribMulAction R'' M₂] [ContinuousConstSMul R'' M₂] [SMulCommClass A R'' M₂] [SMulCommClass R' R'' M₂] : SMulCommClass R' R'' (ContinuousMultilinearMap A M₁ M₂)

instance ContinuousMultilinearMap.instIsScalarTower {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {R' : Type u_1} {R'' : Type u_2} {A : Type u_3} [Monoid R'] [Monoid R''] [Semiring A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [DistribMulAction R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] [DistribMulAction R'' M₂] [ContinuousConstSMul R'' M₂] [SMulCommClass A R'' M₂] [SMul R' R''] [IsScalarTower R' R'' M₂] : IsScalarTower R' R'' (ContinuousMultilinearMap A M₁ M₂)

instance ContinuousMultilinearMap.instIsCentralScalar {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {R' : Type u_1} {A : Type u_3} [Monoid R'] [Semiring A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [DistribMulAction R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] [DistribMulAction R'ᵐᵒᵖ M₂] [IsCentralScalar R' M₂] : IsCentralScalar R' (ContinuousMultilinearMap A M₁ M₂)

instance ContinuousMultilinearMap.instMulAction {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {R' : Type u_1} {A : Type u_3} [Monoid R'] [Semiring A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [DistribMulAction R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] : MulAction R' (ContinuousMultilinearMap A M₁ M₂)

Equations

• ContinuousMultilinearMap.instMulAction = Function.Injective.mulAction ContinuousMultilinearMap.toMultilinearMap ⋯ ⋯

instance ContinuousMultilinearMap.instAdd {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousAdd M₂] : Add (ContinuousMultilinearMap R M₁ M₂)

Equations

• ContinuousMultilinearMap.instAdd = { add := fun (f f' : ContinuousMultilinearMap R M₁ M₂) => { toMultilinearMap := f.toMultilinearMap + f'.toMultilinearMap, cont := ⋯ } }

@[simp]

theorem ContinuousMultilinearMap.add_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f f' : ContinuousMultilinearMap R M₁ M₂) [ContinuousAdd M₂] (m : (i : ι) → M₁ i) : (f + f') m = f m + f' m

@[simp]

theorem ContinuousMultilinearMap.toMultilinearMap_add {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousAdd M₂] (f g : ContinuousMultilinearMap R M₁ M₂) : (f + g).toMultilinearMap = f.toMultilinearMap + g.toMultilinearMap

instance ContinuousMultilinearMap.addCommMonoid {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousAdd M₂] : AddCommMonoid (ContinuousMultilinearMap R M₁ M₂)

Equations

• ContinuousMultilinearMap.addCommMonoid = Function.Injective.addCommMonoid ContinuousMultilinearMap.toMultilinearMap ⋯ ⋯ ⋯ ⋯

def ContinuousMultilinearMap.applyAddHom {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousAdd M₂] (m : (i : ι) → M₁ i) : ContinuousMultilinearMap R M₁ M₂ →+ M₂

Evaluation of a ContinuousMultilinearMap at a vector as an AddMonoidHom.

Equations

• ContinuousMultilinearMap.applyAddHom m = { toFun := fun (f : ContinuousMultilinearMap R M₁ M₂) => f m, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.sum_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousAdd M₂] {α : Type u_1} (f : α → ContinuousMultilinearMap R M₁ M₂) (m : (i : ι) → M₁ i) {s : Finset α} : (∑ a ∈ s, f a) m = ∑ a ∈ s, (f a) m

def ContinuousMultilinearMap.toContinuousLinearMap {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) : M₁ i →L[R] M₂

If f is a continuous multilinear map, then f.toContinuousLinearMap m i is the continuous linear map obtained by fixing all coordinates but i equal to those of m, and varying the i-th coordinate.

Equations

• f.toContinuousLinearMap m i = { toLinearMap := f.toLinearMap m i, cont := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.toContinuousLinearMap_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x : M₁ i) : (f.toContinuousLinearMap m i) x = f (Function.update m i x)

def ContinuousMultilinearMap.prod {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [AddCommMonoid M₃] [(i : ι) → Module R (M₁ i)] [Module R M₂] [Module R M₃] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [TopologicalSpace M₃] (f : ContinuousMultilinearMap R M₁ M₂) (g : ContinuousMultilinearMap R M₁ M₃) : ContinuousMultilinearMap R M₁ (M₂ × M₃)

The cartesian product of two continuous multilinear maps, as a continuous multilinear map.

Equations

• f.prod g = { toMultilinearMap := f.prod g.toMultilinearMap, cont := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.prod_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [AddCommMonoid M₃] [(i : ι) → Module R (M₁ i)] [Module R M₂] [Module R M₃] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [TopologicalSpace M₃] (f : ContinuousMultilinearMap R M₁ M₂) (g : ContinuousMultilinearMap R M₁ M₃) (m : (i : ι) → M₁ i) : (f.prod g) m = (f m, g m)

def ContinuousMultilinearMap.pi {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] {ι' : Type u_1} {M' : ι' → Type u_2} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [(i : ι') → Module R (M' i)] (f : (i : ι') → ContinuousMultilinearMap R M₁ (M' i)) : ContinuousMultilinearMap R M₁ ((i : ι') → M' i)

Combine a family of continuous multilinear maps with the same domain and codomains M' i into a continuous multilinear map taking values in the space of functions ∀ i, M' i.

Equations

• ContinuousMultilinearMap.pi f = { toMultilinearMap := MultilinearMap.pi fun (i : ι') => (f i).toMultilinearMap, cont := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.coe_pi {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] {ι' : Type u_1} {M' : ι' → Type u_2} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [(i : ι') → Module R (M' i)] (f : (i : ι') → ContinuousMultilinearMap R M₁ (M' i)) : ⇑(ContinuousMultilinearMap.pi f) = fun (m : (i : ι) → M₁ i) (j : ι') => (f j) m

theorem ContinuousMultilinearMap.pi_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] {ι' : Type u_1} {M' : ι' → Type u_2} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [(i : ι') → Module R (M' i)] (f : (i : ι') → ContinuousMultilinearMap R M₁ (M' i)) (m : (i : ι) → M₁ i) (j : ι') : (ContinuousMultilinearMap.pi f) m j = (f j) m

def ContinuousMultilinearMap.codRestrict {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) (p : Submodule R M₂) (h : ∀ (v : (i : ι) → M₁ i), f v ∈ p) : ContinuousMultilinearMap R M₁ ↥p

Restrict the codomain of a continuous multilinear map to a submodule.

Equations

• f.codRestrict p h = { toMultilinearMap := f.codRestrict p h, cont := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.codRestrict_toMultilinearMap {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) (p : Submodule R M₂) (h : ∀ (v : (i : ι) → M₁ i), f v ∈ p) : (f.codRestrict p h).toMultilinearMap = f.codRestrict p h

@[simp]

theorem ContinuousMultilinearMap.codRestrict_apply_coe {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) (p : Submodule R M₂) (h : ∀ (v : (i : ι) → M₁ i), f v ∈ p) (v : (i : ι) → M₁ i) : ↑((f.codRestrict p h) v) = f v

def ContinuousMultilinearMap.ofSubsingleton (R : Type u) {ι : Type v} (M₂ : Type w₂) (M₃ : Type w₃) [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] [Subsingleton ι] (i : ι) : (M₂ →L[R] M₃) ≃ ContinuousMultilinearMap R (fun (x : ι) => M₂) M₃

The natural equivalence between continuous linear maps from M₂ to M₃ and continuous 1-multilinear maps from M₂ to M₃.

Equations

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

@[simp]

theorem ContinuousMultilinearMap.ofSubsingleton_apply_apply (R : Type u) {ι : Type v} (M₂ : Type w₂) (M₃ : Type w₃) [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] [Subsingleton ι] (i : ι) (f : M₂ →L[R] M₃) (x : ι → M₂) : ((ContinuousMultilinearMap.ofSubsingleton R M₂ M₃ i) f) x = f (x i)

@[simp]

theorem ContinuousMultilinearMap.ofSubsingleton_symm_apply_apply (R : Type u) {ι : Type v} (M₂ : Type w₂) (M₃ : Type w₃) [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] [Subsingleton ι] (i : ι) (f : ContinuousMultilinearMap R (fun (x : ι) => M₂) M₃) (x : M₂) : ((ContinuousMultilinearMap.ofSubsingleton R M₂ M₃ i).symm f) x = f fun (x_1 : ι) => x

@[simp]

theorem ContinuousMultilinearMap.ofSubsingleton_apply_toMultilinearMap (R : Type u) {ι : Type v} (M₂ : Type w₂) (M₃ : Type w₃) [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] [Subsingleton ι] (i : ι) (f : M₂ →L[R] M₃) : ((ContinuousMultilinearMap.ofSubsingleton R M₂ M₃ i) f).toMultilinearMap = (MultilinearMap.ofSubsingleton R M₂ M₃ i) ↑f

def ContinuousMultilinearMap.constOfIsEmpty (R : Type u) {ι : Type v} (M₁ : ι → Type w₁) {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [IsEmpty ι] (m : M₂) : ContinuousMultilinearMap R M₁ M₂

The constant map is multilinear when ι is empty.

Equations

• ContinuousMultilinearMap.constOfIsEmpty R M₁ m = { toMultilinearMap := MultilinearMap.constOfIsEmpty R M₁ m, cont := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.constOfIsEmpty_toMultilinearMap (R : Type u) {ι : Type v} (M₁ : ι → Type w₁) {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [IsEmpty ι] (m : M₂) : (ContinuousMultilinearMap.constOfIsEmpty R M₁ m).toMultilinearMap = MultilinearMap.constOfIsEmpty R M₁ m

@[simp]

theorem ContinuousMultilinearMap.constOfIsEmpty_apply (R : Type u) {ι : Type v} (M₁ : ι → Type w₁) {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [IsEmpty ι] (m : M₂) (a✝ : (i : ι) → M₁ i) : (ContinuousMultilinearMap.constOfIsEmpty R M₁ m) a✝ = m

def ContinuousMultilinearMap.compContinuousLinearMap {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₁' : ι → Type w₁'} {M₄ : Type w₄} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → AddCommMonoid (M₁' i)] [AddCommMonoid M₄] [(i : ι) → Module R (M₁ i)] [(i : ι) → Module R (M₁' i)] [Module R M₄] [(i : ι) → TopologicalSpace (M₁ i)] [(i : ι) → TopologicalSpace (M₁' i)] [TopologicalSpace M₄] (g : ContinuousMultilinearMap R M₁' M₄) (f : (i : ι) → M₁ i →L[R] M₁' i) : ContinuousMultilinearMap R M₁ M₄

If g is continuous multilinear and f is a collection of continuous linear maps, then g (f₁ m₁, ..., fₙ mₙ) is again a continuous multilinear map, that we call g.compContinuousLinearMap f.

Equations

• g.compContinuousLinearMap f = { toMultilinearMap := g.compLinearMap fun (i : ι) => ↑(f i), cont := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.compContinuousLinearMap_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₁' : ι → Type w₁'} {M₄ : Type w₄} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → AddCommMonoid (M₁' i)] [AddCommMonoid M₄] [(i : ι) → Module R (M₁ i)] [(i : ι) → Module R (M₁' i)] [Module R M₄] [(i : ι) → TopologicalSpace (M₁ i)] [(i : ι) → TopologicalSpace (M₁' i)] [TopologicalSpace M₄] (g : ContinuousMultilinearMap R M₁' M₄) (f : (i : ι) → M₁ i →L[R] M₁' i) (m : (i : ι) → M₁ i) : (g.compContinuousLinearMap f) m = g fun (i : ι) => (f i) (m i)

def ContinuousLinearMap.compContinuousMultilinearMap {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [AddCommMonoid M₃] [(i : ι) → Module R (M₁ i)] [Module R M₂] [Module R M₃] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [TopologicalSpace M₃] (g : M₂ →L[R] M₃) (f : ContinuousMultilinearMap R M₁ M₂) : ContinuousMultilinearMap R M₁ M₃

Composing a continuous multilinear map with a continuous linear map gives again a continuous multilinear map.

Equations

• g.compContinuousMultilinearMap f = { toMultilinearMap := (↑g).compMultilinearMap f.toMultilinearMap, cont := ⋯ }

@[simp]

theorem ContinuousLinearMap.compContinuousMultilinearMap_coe {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [AddCommMonoid M₃] [(i : ι) → Module R (M₁ i)] [Module R M₂] [Module R M₃] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [TopologicalSpace M₃] (g : M₂ →L[R] M₃) (f : ContinuousMultilinearMap R M₁ M₂) : ⇑(g.compContinuousMultilinearMap f) = ⇑g ∘ ⇑f

def ContinuousMultilinearMap.piEquiv {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] {ι' : Type u_1} {M' : ι' → Type u_2} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [(i : ι') → Module R (M' i)] : ((i : ι') → ContinuousMultilinearMap R M₁ (M' i)) ≃ ContinuousMultilinearMap R M₁ ((i : ι') → M' i)

ContinuousMultilinearMap.pi as an Equiv.

Equations

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

@[simp]

theorem ContinuousMultilinearMap.piEquiv_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] {ι' : Type u_1} {M' : ι' → Type u_2} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [(i : ι') → Module R (M' i)] (f : (i : ι') → ContinuousMultilinearMap R M₁ (M' i)) : ContinuousMultilinearMap.piEquiv f = ContinuousMultilinearMap.pi f

@[simp]

theorem ContinuousMultilinearMap.piEquiv_symm_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] {ι' : Type u_1} {M' : ι' → Type u_2} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [(i : ι') → Module R (M' i)] (f : ContinuousMultilinearMap R M₁ ((i : ι') → M' i)) (i : ι') : ContinuousMultilinearMap.piEquiv.symm f i = (ContinuousLinearMap.proj i).compContinuousMultilinearMap f

def ContinuousMultilinearMap.domDomCongr {R : Type u} {ι : Type v} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] {ι' : Type u_1} (e : ι ≃ ι') (f : ContinuousMultilinearMap R (fun (x : ι) => M₂) M₃) : ContinuousMultilinearMap R (fun (x : ι') => M₂) M₃

An equivalence of the index set defines an equivalence between the spaces of continuous multilinear maps. This is the forward map of this equivalence.

Equations

• ContinuousMultilinearMap.domDomCongr e f = { toMultilinearMap := MultilinearMap.domDomCongr e f.toMultilinearMap, cont := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.domDomCongr_apply {R : Type u} {ι : Type v} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] {ι' : Type u_1} (e : ι ≃ ι') (f : ContinuousMultilinearMap R (fun (x : ι) => M₂) M₃) (v : ι' → M₂) : (ContinuousMultilinearMap.domDomCongr e f) v = f fun (i : ι) => v (e i)

@[simp]

theorem ContinuousMultilinearMap.domDomCongr_toMultilinearMap {R : Type u} {ι : Type v} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] {ι' : Type u_1} (e : ι ≃ ι') (f : ContinuousMultilinearMap R (fun (x : ι) => M₂) M₃) : (ContinuousMultilinearMap.domDomCongr e f).toMultilinearMap = MultilinearMap.domDomCongr e f.toMultilinearMap

def ContinuousMultilinearMap.domDomCongrEquiv {R : Type u} {ι : Type v} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] {ι' : Type u_1} (e : ι ≃ ι') : ContinuousMultilinearMap R (fun (x : ι) => M₂) M₃ ≃ ContinuousMultilinearMap R (fun (x : ι') => M₂) M₃

An equivalence of the index set defines an equivalence between the spaces of continuous multilinear maps. In case of normed spaces, this is a linear isometric equivalence, see ContinuousMultilinearMap.domDomCongrₗᵢ.

Equations

• ContinuousMultilinearMap.domDomCongrEquiv e = { toFun := ContinuousMultilinearMap.domDomCongr e, invFun := ContinuousMultilinearMap.domDomCongr e.symm, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.domDomCongrEquiv_symm_apply {R : Type u} {ι : Type v} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] {ι' : Type u_1} (e : ι ≃ ι') (f : ContinuousMultilinearMap R (fun (x : ι') => M₂) M₃) : (ContinuousMultilinearMap.domDomCongrEquiv e).symm f = ContinuousMultilinearMap.domDomCongr e.symm f

@[simp]

theorem ContinuousMultilinearMap.domDomCongrEquiv_apply {R : Type u} {ι : Type v} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] {ι' : Type u_1} (e : ι ≃ ι') (f : ContinuousMultilinearMap R (fun (x : ι) => M₂) M₃) : (ContinuousMultilinearMap.domDomCongrEquiv e) f = ContinuousMultilinearMap.domDomCongr e f

def ContinuousMultilinearMap.linearDeriv {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [ContinuousAdd M₂] [DecidableEq ι] [Fintype ι] (x : (i : ι) → M₁ i) : ((i : ι) → M₁ i) →L[R] M₂

The derivative of a continuous multilinear map, as a continuous linear map from ∀ i, M₁ i to M₂; see ContinuousMultilinearMap.hasFDerivAt.

Equations

• f.linearDeriv x = ∑ i : ι, (f.toContinuousLinearMap x i).comp (ContinuousLinearMap.proj i)

@[simp]

theorem ContinuousMultilinearMap.linearDeriv_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [ContinuousAdd M₂] [DecidableEq ι] [Fintype ι] (x y : (i : ι) → M₁ i) : (f.linearDeriv x) y = ∑ i : ι, f (Function.update x i (y i))

theorem ContinuousMultilinearMap.cons_add {R : Type u} {n : ℕ} {M : Fin n.succ → Type w} {M₂ : Type w₂} [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [AddCommMonoid M₂] [(i : Fin n.succ) → Module R (M i)] [Module R M₂] [(i : Fin n.succ) → TopologicalSpace (M i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M M₂) (m : (i : Fin n) → M i.succ) (x y : M 0) : f (Fin.cons (x + y) m) = f (Fin.cons x m) + f (Fin.cons y m)

In the specific case of continuous multilinear maps on spaces indexed by Fin (n+1), where one can build an element of (i : Fin (n+1)) → M i using cons, one can express directly the additivity of a multilinear map along the first variable.

theorem ContinuousMultilinearMap.cons_smul {R : Type u} {n : ℕ} {M : Fin n.succ → Type w} {M₂ : Type w₂} [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [AddCommMonoid M₂] [(i : Fin n.succ) → Module R (M i)] [Module R M₂] [(i : Fin n.succ) → TopologicalSpace (M i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M M₂) (m : (i : Fin n) → M i.succ) (c : R) (x : M 0) : f (Fin.cons (c • x) m) = c • f (Fin.cons x m)

In the specific case of continuous multilinear maps on spaces indexed by Fin (n+1), where one can build an element of (i : Fin (n+1)) → M i using cons, one can express directly the multiplicativity of a multilinear map along the first variable.

theorem ContinuousMultilinearMap.map_piecewise_add {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (m m' : (i : ι) → M₁ i) (t : Finset ι) : f (t.piecewise (m + m') m') = ∑ s ∈ t.powerset, f (s.piecewise m m')

theorem ContinuousMultilinearMap.map_add_univ {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] [Fintype ι] (m m' : (i : ι) → M₁ i) : f (m + m') = ∑ s : Finset ι, f (s.piecewise m m')

Additivity of a continuous multilinear map along all coordinates at the same time, writing f (m + m') as the sum of f (s.piecewise m m') over all sets s.

theorem ContinuousMultilinearMap.map_sum_finset {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) {α : ι → Type u_1} [Fintype ι] (g : (i : ι) → α i → M₁ i) (A : (i : ι) → Finset (α i)) [DecidableEq ι] : (f fun (i : ι) => ∑ j ∈ A i, g i j) = ∑ r ∈ Fintype.piFinset A, f fun (i : ι) => g i (r i)

If f is continuous multilinear, then f (Σ_{j₁ ∈ A₁} g₁ j₁, ..., Σ_{jₙ ∈ Aₙ} gₙ jₙ) is the sum of f (g₁ (r 1), ..., gₙ (r n)) where r ranges over all functions with r 1 ∈ A₁, ..., r n ∈ Aₙ. This follows from multilinearity by expanding successively with respect to each coordinate.

theorem ContinuousMultilinearMap.map_sum {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) {α : ι → Type u_1} [Fintype ι] (g : (i : ι) → α i → M₁ i) [DecidableEq ι] [(i : ι) → Fintype (α i)] : (f fun (i : ι) => ∑ j : α i, g i j) = ∑ r : (i : ι) → α i, f fun (i : ι) => g i (r i)

If f is continuous multilinear, then f (Σ_{j₁} g₁ j₁, ..., Σ_{jₙ} gₙ jₙ) is the sum of f (g₁ (r 1), ..., gₙ (r n)) where r ranges over all functions r. This follows from multilinearity by expanding successively with respect to each coordinate.

def ContinuousMultilinearMap.restrictScalars (R : Type u) {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {A : Type u_1} [Semiring A] [SMul R A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [∀ (i : ι), IsScalarTower R A (M₁ i)] [IsScalarTower R A M₂] (f : ContinuousMultilinearMap A M₁ M₂) : ContinuousMultilinearMap R M₁ M₂

Reinterpret an A-multilinear map as an R-multilinear map, if A is an algebra over R and their actions on all involved modules agree with the action of R on A.

Equations

• ContinuousMultilinearMap.restrictScalars R f = { toMultilinearMap := MultilinearMap.restrictScalars R f.toMultilinearMap, cont := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.coe_restrictScalars (R : Type u) {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {A : Type u_1} [Semiring A] [SMul R A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [∀ (i : ι), IsScalarTower R A (M₁ i)] [IsScalarTower R A M₂] (f : ContinuousMultilinearMap A M₁ M₂) : ⇑(ContinuousMultilinearMap.restrictScalars R f) = ⇑f

@[simp]

theorem ContinuousMultilinearMap.map_update_sub {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Ring R] [(i : ι) → AddCommGroup (M₁ i)] [AddCommGroup M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i) : f (Function.update m i (x - y)) = f (Function.update m i x) - f (Function.update m i y)

@[deprecated ContinuousMultilinearMap.map_update_sub]

theorem ContinuousMultilinearMap.map_sub {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Ring R] [(i : ι) → AddCommGroup (M₁ i)] [AddCommGroup M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i) : f (Function.update m i (x - y)) = f (Function.update m i x) - f (Function.update m i y)

Alias of ContinuousMultilinearMap.map_update_sub.

instance ContinuousMultilinearMap.instNeg {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Ring R] [(i : ι) → AddCommGroup (M₁ i)] [AddCommGroup M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [TopologicalAddGroup M₂] : Neg (ContinuousMultilinearMap R M₁ M₂)

Equations

• ContinuousMultilinearMap.instNeg = { neg := fun (f : ContinuousMultilinearMap R M₁ M₂) => let __src := -f.toMultilinearMap; { toMultilinearMap := __src, cont := ⋯ } }

@[simp]

theorem ContinuousMultilinearMap.neg_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Ring R] [(i : ι) → AddCommGroup (M₁ i)] [AddCommGroup M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [TopologicalAddGroup M₂] (m : (i : ι) → M₁ i) : (-f) m = -f m

instance ContinuousMultilinearMap.instSub {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Ring R] [(i : ι) → AddCommGroup (M₁ i)] [AddCommGroup M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [TopologicalAddGroup M₂] : Sub (ContinuousMultilinearMap R M₁ M₂)

Equations

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

@[simp]

theorem ContinuousMultilinearMap.sub_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Ring R] [(i : ι) → AddCommGroup (M₁ i)] [AddCommGroup M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f f' : ContinuousMultilinearMap R M₁ M₂) [TopologicalAddGroup M₂] (m : (i : ι) → M₁ i) : (f - f') m = f m - f' m

instance ContinuousMultilinearMap.instAddCommGroup {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Ring R] [(i : ι) → AddCommGroup (M₁ i)] [AddCommGroup M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [TopologicalAddGroup M₂] : AddCommGroup (ContinuousMultilinearMap R M₁ M₂)

Equations

• ContinuousMultilinearMap.instAddCommGroup = Function.Injective.addCommGroup ContinuousMultilinearMap.toMultilinearMap ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

theorem ContinuousMultilinearMap.map_piecewise_smul {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [CommSemiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (c : ι → R) (m : (i : ι) → M₁ i) (s : Finset ι) : f (s.piecewise (fun (i : ι) => c i • m i) m) = (∏ i ∈ s, c i) • f m

theorem ContinuousMultilinearMap.map_smul_univ {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [CommSemiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [Fintype ι] (c : ι → R) (m : (i : ι) → M₁ i) : (f fun (i : ι) => c i • m i) = (∏ i : ι, c i) • f m

Multiplicativity of a continuous multilinear map along all coordinates at the same time, writing f (fun i ↦ c i • m i) as (∏ i, c i) • f m.

instance ContinuousMultilinearMap.instDistribMulAction {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} {R' : Type u_1} {A : Type u_3} [Monoid R'] [Semiring A] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [(i : ι) → Module A (M₁ i)] [Module A M₂] [DistribMulAction R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] [ContinuousAdd M₂] : DistribMulAction R' (ContinuousMultilinearMap A M₁ M₂)

Equations

• ContinuousMultilinearMap.instDistribMulAction = Function.Injective.distribMulAction { toFun := ContinuousMultilinearMap.toMultilinearMap, map_zero' := ⋯, map_add' := ⋯ } ⋯ ⋯

instance ContinuousMultilinearMap.instModule {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} {R' : Type u_1} {A : Type u_2} [Semiring R'] [Semiring A] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousAdd M₂] [(i : ι) → Module A (M₁ i)] [Module A M₂] [Module R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] : Module R' (ContinuousMultilinearMap A M₁ M₂)

The space of continuous multilinear maps over an algebra over R is a module over R, for the pointwise addition and scalar multiplication.

Equations

• ContinuousMultilinearMap.instModule = Function.Injective.module R' { toFun := ContinuousMultilinearMap.toMultilinearMap, map_zero' := ⋯, map_add' := ⋯ } ⋯ ⋯

def ContinuousMultilinearMap.toMultilinearMapLinear {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} {R' : Type u_1} {A : Type u_2} [Semiring R'] [Semiring A] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousAdd M₂] [(i : ι) → Module A (M₁ i)] [Module A M₂] [Module R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] : ContinuousMultilinearMap A M₁ M₂ →ₗ[R'] MultilinearMap A M₁ M₂

Linear map version of the map toMultilinearMap associating to a continuous multilinear map the corresponding multilinear map.

Equations

• ContinuousMultilinearMap.toMultilinearMapLinear = { toFun := ContinuousMultilinearMap.toMultilinearMap, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.toMultilinearMapLinear_apply {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} {R' : Type u_1} {A : Type u_2} [Semiring R'] [Semiring A] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousAdd M₂] [(i : ι) → Module A (M₁ i)] [Module A M₂] [Module R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] (self : ContinuousMultilinearMap A M₁ M₂) : ContinuousMultilinearMap.toMultilinearMapLinear self = self.toMultilinearMap

def ContinuousMultilinearMap.piLinearEquiv {ι : Type v} {M₁ : ι → Type w₁} {R' : Type u_1} {A : Type u_2} [Semiring R'] [Semiring A] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] [(i : ι) → Module A (M₁ i)] {ι' : Type u_3} {M' : ι' → Type u_4} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [∀ (i : ι'), ContinuousAdd (M' i)] [(i : ι') → Module R' (M' i)] [(i : ι') → Module A (M' i)] [∀ (i : ι'), SMulCommClass A R' (M' i)] [∀ (i : ι'), ContinuousConstSMul R' (M' i)] : ((i : ι') → ContinuousMultilinearMap A M₁ (M' i)) ≃ₗ[R'] ContinuousMultilinearMap A M₁ ((i : ι') → M' i)

ContinuousMultilinearMap.pi as a LinearEquiv.

Equations

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

@[simp]

theorem ContinuousMultilinearMap.piLinearEquiv_symm_apply {ι : Type v} {M₁ : ι → Type w₁} {R' : Type u_1} {A : Type u_2} [Semiring R'] [Semiring A] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] [(i : ι) → Module A (M₁ i)] {ι' : Type u_3} {M' : ι' → Type u_4} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [∀ (i : ι'), ContinuousAdd (M' i)] [(i : ι') → Module R' (M' i)] [(i : ι') → Module A (M' i)] [∀ (i : ι'), SMulCommClass A R' (M' i)] [∀ (i : ι'), ContinuousConstSMul R' (M' i)] (a✝ : ContinuousMultilinearMap A M₁ ((i : ι') → M' i)) (i : ι') : ContinuousMultilinearMap.piLinearEquiv.symm a✝ i = (ContinuousLinearMap.proj i).compContinuousMultilinearMap a✝

@[simp]

theorem ContinuousMultilinearMap.piLinearEquiv_apply {ι : Type v} {M₁ : ι → Type w₁} {R' : Type u_1} {A : Type u_2} [Semiring R'] [Semiring A] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] [(i : ι) → Module A (M₁ i)] {ι' : Type u_3} {M' : ι' → Type u_4} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [∀ (i : ι'), ContinuousAdd (M' i)] [(i : ι') → Module R' (M' i)] [(i : ι') → Module A (M' i)] [∀ (i : ι'), SMulCommClass A R' (M' i)] [∀ (i : ι'), ContinuousConstSMul R' (M' i)] (a✝ : (i : ι') → ContinuousMultilinearMap A M₁ (M' i)) : ContinuousMultilinearMap.piLinearEquiv a✝ = ContinuousMultilinearMap.pi a✝

def ContinuousMultilinearMap.mkPiAlgebra (R : Type u) (ι : Type v) (A : Type u_1) [Fintype ι] [CommSemiring R] [CommSemiring A] [Algebra R A] [TopologicalSpace A] [ContinuousMul A] : ContinuousMultilinearMap R (fun (x : ι) => A) A

The continuous multilinear map on A^ι, where A is a normed commutative algebra over 𝕜, associating to m the product of all the m i.

See also ContinuousMultilinearMap.mkPiAlgebraFin.

Equations

• ContinuousMultilinearMap.mkPiAlgebra R ι A = { toMultilinearMap := MultilinearMap.mkPiAlgebra R ι A, cont := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.mkPiAlgebra_apply (R : Type u) (ι : Type v) (A : Type u_1) [Fintype ι] [CommSemiring R] [CommSemiring A] [Algebra R A] [TopologicalSpace A] [ContinuousMul A] (m : ι → A) : (ContinuousMultilinearMap.mkPiAlgebra R ι A) m = ∏ i : ι, m i

def ContinuousMultilinearMap.mkPiAlgebraFin (R : Type u) (n : ℕ) (A : Type u_1) [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [ContinuousMul A] : ContinuousMultilinearMap R (fun (i : Fin n) => A) A

The continuous multilinear map on A^n, where A is a normed algebra over 𝕜, associating to m the product of all the m i.

See also: ContinuousMultilinearMap.mkPiAlgebra.

Equations

• ContinuousMultilinearMap.mkPiAlgebraFin R n A = { toMultilinearMap := MultilinearMap.mkPiAlgebraFin R n A, cont := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.mkPiAlgebraFin_apply {R : Type u} {n : ℕ} {A : Type u_1} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [ContinuousMul A] (m : Fin n → A) : (ContinuousMultilinearMap.mkPiAlgebraFin R n A) m = (List.ofFn m).prod

def ContinuousMultilinearMap.smulRight {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [CommSemiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [TopologicalSpace R] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousSMul R M₂] (f : ContinuousMultilinearMap R M₁ R) (z : M₂) : ContinuousMultilinearMap R M₁ M₂

Given a continuous R-multilinear map f taking values in R, f.smulRight z is the continuous multilinear map sending m to f m • z.

Equations

• f.smulRight z = { toMultilinearMap := f.smulRight z, cont := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.smulRight_toMultilinearMap {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [CommSemiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [TopologicalSpace R] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousSMul R M₂] (f : ContinuousMultilinearMap R M₁ R) (z : M₂) : (f.smulRight z).toMultilinearMap = f.smulRight z

@[simp]

theorem ContinuousMultilinearMap.smulRight_apply {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [CommSemiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [TopologicalSpace R] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousSMul R M₂] (f : ContinuousMultilinearMap R M₁ R) (z : M₂) (a✝ : (i : ι) → M₁ i) : (f.smulRight z) a✝ = f a✝ • z

def ContinuousMultilinearMap.mkPiRing (R : Type u) (ι : Type v) {M : Type u_1} [Fintype ι] [CommRing R] [AddCommMonoid M] [Module R M] [TopologicalSpace R] [TopologicalSpace M] [ContinuousMul R] [ContinuousSMul R M] (z : M) : ContinuousMultilinearMap R (fun (x : ι) => R) M

The canonical continuous multilinear map on R^ι, associating to m the product of all the m i (multiplied by a fixed reference element z in the target module)

Equations

• ContinuousMultilinearMap.mkPiRing R ι z = (ContinuousMultilinearMap.mkPiAlgebra R ι R).smulRight z

@[simp]

theorem ContinuousMultilinearMap.mkPiRing_apply {R : Type u} {ι : Type v} {M : Type u_1} [Fintype ι] [CommRing R] [AddCommMonoid M] [Module R M] [TopologicalSpace R] [TopologicalSpace M] [ContinuousMul R] [ContinuousSMul R M] (z : M) (m : ι → R) : (ContinuousMultilinearMap.mkPiRing R ι z) m = (∏ i : ι, m i) • z

theorem ContinuousMultilinearMap.mkPiRing_apply_one_eq_self {R : Type u} {ι : Type v} {M : Type u_1} [Fintype ι] [CommRing R] [AddCommMonoid M] [Module R M] [TopologicalSpace R] [TopologicalSpace M] [ContinuousMul R] [ContinuousSMul R M] (f : ContinuousMultilinearMap R (fun (x : ι) => R) M) : ContinuousMultilinearMap.mkPiRing R ι (f fun (x : ι) => 1) = f

theorem ContinuousMultilinearMap.mkPiRing_eq_iff {R : Type u} {ι : Type v} {M : Type u_1} [Fintype ι] [CommRing R] [AddCommMonoid M] [Module R M] [TopologicalSpace R] [TopologicalSpace M] [ContinuousMul R] [ContinuousSMul R M] {z₁ z₂ : M} : ContinuousMultilinearMap.mkPiRing R ι z₁ = ContinuousMultilinearMap.mkPiRing R ι z₂ ↔ z₁ = z₂

theorem ContinuousMultilinearMap.mkPiRing_zero {R : Type u} {ι : Type v} {M : Type u_1} [Fintype ι] [CommRing R] [AddCommMonoid M] [Module R M] [TopologicalSpace R] [TopologicalSpace M] [ContinuousMul R] [ContinuousSMul R M] : ContinuousMultilinearMap.mkPiRing R ι 0 = 0

theorem ContinuousMultilinearMap.mkPiRing_eq_zero_iff {R : Type u} {ι : Type v} {M : Type u_1} [Fintype ι] [CommRing R] [AddCommMonoid M] [Module R M] [TopologicalSpace R] [TopologicalSpace M] [ContinuousMul R] [ContinuousSMul R M] (z : M) : ContinuousMultilinearMap.mkPiRing R ι z = 0 ↔ z = 0