fact
stringlengths 6
3.84k
| type
stringclasses 11
values | library
stringclasses 32
values | imports
listlengths 1
14
| filename
stringlengths 20
95
| symbolic_name
stringlengths 1
90
| docstring
stringlengths 7
20k
⌀ |
|---|---|---|---|---|---|---|
sub : Sub (M →SL[σ₁₂] M₂) :=
⟨fun f g => ⟨f - g, f.2.sub g.2⟩⟩
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
sub
| null |
addCommGroup : AddCommGroup (M →SL[σ₁₂] M₂) where
__ := ContinuousLinearMap.addCommMonoid
neg := (-·)
sub := (· - ·)
sub_eq_add_neg _ _ := by ext; apply sub_eq_add_neg
nsmul := (· • ·)
zsmul := (· • ·)
zsmul_zero' f := by ext; simp
zsmul_succ' n f := by ext; simp [add_smul, add_comm]
zsmul_neg' n f := by ext; simp [add_smul]
neg_add_cancel _ := by ext; apply neg_add_cancel
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
addCommGroup
| null |
sub_apply (f g : M →SL[σ₁₂] M₂) (x : M) : (f - g) x = f x - g x :=
rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
sub_apply
| null |
coe_sub (f g : M →SL[σ₁₂] M₂) : (↑(f - g) : M →ₛₗ[σ₁₂] M₂) = f - g :=
rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
coe_sub
| null |
coe_sub' (f g : M →SL[σ₁₂] M₂) : ⇑(f - g) = f - g :=
rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
coe_sub'
| null |
toContinuousAddMonoidHom_sub (f g : M →SL[σ₁₂] M₂) :
↑(f - g) = (f - g : ContinuousAddMonoidHom M M₂) := rfl
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
toContinuousAddMonoidHom_sub
| null |
@[simp]
comp_neg [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [IsTopologicalAddGroup M₂]
[IsTopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) :
g.comp (-f) = -g.comp f := by
ext x
simp
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
comp_neg
| null |
neg_comp [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [IsTopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃)
(f : M →SL[σ₁₂] M₂) : (-g).comp f = -g.comp f := by
ext
simp
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
neg_comp
| null |
comp_sub [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [IsTopologicalAddGroup M₂]
[IsTopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f₁ f₂ : M →SL[σ₁₂] M₂) :
g.comp (f₁ - f₂) = g.comp f₁ - g.comp f₂ := by
ext
simp
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
comp_sub
| null |
sub_comp [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [IsTopologicalAddGroup M₃] (g₁ g₂ : M₂ →SL[σ₂₃] M₃)
(f : M →SL[σ₁₂] M₂) : (g₁ - g₂).comp f = g₁.comp f - g₂.comp f := by
ext
simp
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
sub_comp
| null |
ring [IsTopologicalAddGroup M] : Ring (M →L[R] M) where
__ := ContinuousLinearMap.semiring
__ := ContinuousLinearMap.addCommGroup
intCast z := z • (1 : M →L[R] M)
intCast_ofNat := natCast_zsmul _
intCast_negSucc := negSucc_zsmul _
@[simp]
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
ring
| null |
intCast_apply [IsTopologicalAddGroup M] (z : ℤ) (m : M) : (↑z : M →L[R] M) m = z • m :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
intCast_apply
| null |
smulRight_one_pow [TopologicalSpace R] [IsTopologicalRing R] (c : R) (n : ℕ) :
smulRight (1 : R →L[R] R) c ^ n = smulRight (1 : R →L[R] R) (c ^ n) := by
induction n with
| zero => ext; simp
| succ n ihn => rw [pow_succ, ihn, mul_def, smulRight_comp, smul_eq_mul, pow_succ']
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
smulRight_one_pow
| null |
projKerOfRightInverse [IsTopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M)
(h : Function.RightInverse f₂ f₁) : M →L[R] LinearMap.ker f₁ :=
(id R M - f₂.comp f₁).codRestrict (LinearMap.ker f₁) fun x => by simp [h (f₁ x)]
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
projKerOfRightInverse
|
Given a right inverse `f₂ : M₂ →L[R] M` to `f₁ : M →L[R] M₂`,
`projKerOfRightInverse f₁ f₂ h` is the projection `M →L[R] LinearMap.ker f₁` along
`LinearMap.range f₂`.
|
coe_projKerOfRightInverse_apply [IsTopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂)
(f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (x : M) :
(f₁.projKerOfRightInverse f₂ h x : M) = x - f₂ (f₁ x) :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
coe_projKerOfRightInverse_apply
| null |
projKerOfRightInverse_apply_idem [IsTopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂)
(f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (x : LinearMap.ker f₁) :
f₁.projKerOfRightInverse f₂ h x = x := by
ext1
simp
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
projKerOfRightInverse_apply_idem
| null |
projKerOfRightInverse_comp_inv [IsTopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂)
(f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (y : M₂) :
f₁.projKerOfRightInverse f₂ h (f₂ y) = 0 :=
Subtype.ext_iff.2 <| by simp [h y]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
projKerOfRightInverse_comp_inv
| null |
protected isOpenMap_of_ne_zero [TopologicalSpace R] [DivisionRing R] [ContinuousSub R]
[AddCommGroup M] [TopologicalSpace M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M]
(f : StrongDual R M) (hf : f ≠ 0) : IsOpenMap f :=
let ⟨x, hx⟩ := exists_ne_zero hf
IsOpenMap.of_sections fun y =>
⟨fun a => y + (a - f y) • (f x)⁻¹ • x, Continuous.continuousAt <| by fun_prop, by simp,
fun a => by simp [hx]⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
isOpenMap_of_ne_zero
|
A nonzero continuous linear functional is open.
|
@[simp]
smul_comp (c : S₃) (h : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) :
(c • h).comp f = c • h.comp f :=
rfl
variable [DistribMulAction S₃ M₂] [ContinuousConstSMul S₃ M₂] [SMulCommClass R₂ S₃ M₂]
variable [DistribMulAction S N₂] [ContinuousConstSMul S N₂] [SMulCommClass R S N₂]
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
smul_comp
| null |
comp_smul [LinearMap.CompatibleSMul N₂ N₃ S R] (hₗ : N₂ →L[R] N₃) (c : S)
(fₗ : M →L[R] N₂) : hₗ.comp (c • fₗ) = c • hₗ.comp fₗ := by
ext x
exact hₗ.map_smul_of_tower c (fₗ x)
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
comp_smul
| null |
comp_smulₛₗ [SMulCommClass R₂ R₂ M₂] [SMulCommClass R₃ R₃ M₃] [ContinuousConstSMul R₂ M₂]
[ContinuousConstSMul R₃ M₃] (h : M₂ →SL[σ₂₃] M₃) (c : R₂) (f : M →SL[σ₁₂] M₂) :
h.comp (c • f) = σ₂₃ c • h.comp f := by
ext x
simp only [coe_smul', coe_comp', Function.comp_apply, Pi.smul_apply,
ContinuousLinearMap.map_smulₛₗ]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
comp_smulₛₗ
| null |
distribMulAction [ContinuousAdd M₂] : DistribMulAction S₃ (M →SL[σ₁₂] M₂) where
smul_add a f g := ext fun x => smul_add a (f x) (g x)
smul_zero a := ext fun _ => smul_zero a
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
distribMulAction
| null |
module : Module S₃ (M →SL[σ₁₃] M₃) where
zero_smul _ := ext fun _ => zero_smul S₃ _
add_smul _ _ _ := ext fun _ => add_smul _ _ _
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
module
| null |
isCentralScalar [Module S₃ᵐᵒᵖ M₃] [IsCentralScalar S₃ M₃] :
IsCentralScalar S₃ (M →SL[σ₁₃] M₃) where
op_smul_eq_smul _ _ := ext fun _ => op_smul_eq_smul _ _
variable (S) [ContinuousAdd N₃]
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
isCentralScalar
| null |
@[simps]
coeLM : (M →L[R] N₃) →ₗ[S] M →ₗ[R] N₃ where
toFun := (↑)
map_add' f g := coe_add f g
map_smul' c f := coe_smul c f
variable {S} (σ₁₃)
|
def
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
coeLM
|
The coercion from `M →L[R] M₂` to `M →ₗ[R] M₂`, as a linear map.
|
@[simps]
coeLMₛₗ : (M →SL[σ₁₃] M₃) →ₗ[S₃] M →ₛₗ[σ₁₃] M₃ where
toFun := (↑)
map_add' f g := coe_add f g
map_smul' c f := coe_smul c f
|
def
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
coeLMₛₗ
|
The coercion from `M →SL[σ] M₂` to `M →ₛₗ[σ] M₂`, as a linear map.
|
smulRightₗ (c : M →L[R] S) : M₂ →ₗ[T] M →L[R] M₂ where
toFun := c.smulRight
map_add' x y := by
ext e
apply smul_add (c e)
map_smul' a x := by
ext e
dsimp
apply smul_comm
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
smulRightₗ
|
Given `c : E →L[R] S`, `c.smulRightₗ` is the linear map from `F` to `E →L[R] F`
sending `f` to `fun e => c e • f`. See also `ContinuousLinearMap.smulRightL`.
|
coe_smulRightₗ (c : M →L[R] S) : ⇑(smulRightₗ c : M₂ →ₗ[T] M →L[R] M₂) = c.smulRight :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
coe_smulRightₗ
| null |
algebra : Algebra R (M₂ →L[R] M₂) :=
Algebra.ofModule smul_comp fun _ _ _ => comp_smul _ _ _
@[simp] theorem algebraMap_apply (r : R) (m : M₂) : algebraMap R (M₂ →L[R] M₂) r m = r • m := rfl
|
instance
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
algebra
| null |
restrictScalars (f : M₁ →L[A] M₂) : M₁ →L[R] M₂ :=
⟨(f : M₁ →ₗ[A] M₂).restrictScalars R, f.continuous⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
restrictScalars
|
If `A` is an `R`-algebra, then a continuous `A`-linear map can be interpreted as a continuous
`R`-linear map. We assume `LinearMap.CompatibleSMul M₁ M₂ R A` to match assumptions of
`LinearMap.map_smul_of_tower`.
|
coe_restrictScalars (f : M₁ →L[A] M₂) :
(f.restrictScalars R : M₁ →ₗ[R] M₂) = (f : M₁ →ₗ[A] M₂).restrictScalars R := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
coe_restrictScalars
| null |
coe_restrictScalars' (f : M₁ →L[A] M₂) : ⇑(f.restrictScalars R) = f := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
coe_restrictScalars'
| null |
toContinuousAddMonoidHom_restrictScalars (f : M₁ →L[A] M₂) :
↑(f.restrictScalars R) = (f : ContinuousAddMonoidHom M₁ M₂) := rfl
@[simp] lemma restrictScalars_zero : (0 : M₁ →L[A] M₂).restrictScalars R = 0 := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
toContinuousAddMonoidHom_restrictScalars
| null |
restrictScalars_add [ContinuousAdd M₂] (f g : M₁ →L[A] M₂) :
(f + g).restrictScalars R = f.restrictScalars R + g.restrictScalars R := rfl
variable [Module S M₂] [ContinuousConstSMul S M₂] [SMulCommClass A S M₂] [SMulCommClass R S M₂]
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
restrictScalars_add
| null |
restrictScalars_smul (c : S) (f : M₁ →L[A] M₂) :
(c • f).restrictScalars R = c • f.restrictScalars R :=
rfl
variable [ContinuousAdd M₂]
variable (A R S M₁ M₂) in
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
restrictScalars_smul
| null |
restrictScalarsₗ : (M₁ →L[A] M₂) →ₗ[S] M₁ →L[R] M₂ where
toFun := restrictScalars R
map_add' := restrictScalars_add
map_smul' := restrictScalars_smul
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
restrictScalarsₗ
|
`ContinuousLinearMap.restrictScalars` as a `LinearMap`. See also
`ContinuousLinearMap.restrictScalarsL`.
|
coe_restrictScalarsₗ : ⇑(restrictScalarsₗ A M₁ M₂ R S) = restrictScalars R := rfl
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
coe_restrictScalarsₗ
| null |
@[simp]
restrictScalars_sub (f g : M₁ →L[A] M₂) :
(f - g).restrictScalars R = f.restrictScalars R - g.restrictScalars R := rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
restrictScalars_sub
| null |
restrictScalars_neg (f : M₁ →L[A] M₂) : (-f).restrictScalars R = -f.restrictScalars R := rfl
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
restrictScalars_neg
| null |
ClosedComplemented (p : Submodule R M) : Prop :=
∃ f : M →L[R] p, ∀ x : p, f x = x
|
def
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
ClosedComplemented
|
A submodule `p` is called *complemented* if there exists a continuous projection `M →ₗ[R] p`.
|
ClosedComplemented.exists_isClosed_isCompl {p : Submodule R M} [T1Space p]
(h : ClosedComplemented p) :
∃ q : Submodule R M, IsClosed (q : Set M) ∧ IsCompl p q :=
Exists.elim h fun f hf => ⟨ker f, isClosed_ker f, LinearMap.isCompl_of_proj hf⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
ClosedComplemented.exists_isClosed_isCompl
| null |
protected ClosedComplemented.isClosed [IsTopologicalAddGroup M] [T1Space M]
{p : Submodule R M} (h : ClosedComplemented p) : IsClosed (p : Set M) := by
rcases h with ⟨f, hf⟩
have : ker (id R M - p.subtypeL.comp f) = p := LinearMap.ker_id_sub_eq_of_proj hf
exact this ▸ isClosed_ker _
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
ClosedComplemented.isClosed
| null |
closedComplemented_bot : ClosedComplemented (⊥ : Submodule R M) :=
⟨0, fun x => by simp only [zero_apply, eq_zero_of_bot_submodule x]⟩
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
closedComplemented_bot
| null |
closedComplemented_top : ClosedComplemented (⊤ : Submodule R M) :=
⟨(id R M).codRestrict ⊤ fun _x => trivial, fun x => Subtype.ext_iff.2 <| by simp⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
closedComplemented_top
| null |
ContinuousLinearMap.closedComplemented_ker_of_rightInverse {R : Type*} [Ring R]
{M : Type*} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type*} [TopologicalSpace M₂]
[AddCommGroup M₂] [Module R M] [Module R M₂] [IsTopologicalAddGroup M] (f₁ : M →L[R] M₂)
(f₂ : M₂ →L[R] M) (h : Function.RightInverse f₂ f₁) : (ker f₁).ClosedComplemented :=
⟨f₁.projKerOfRightInverse f₂ h, f₁.projKerOfRightInverse_apply_idem f₂ h⟩
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
ContinuousLinearMap.closedComplemented_ker_of_rightInverse
| null |
@[grind =]
isIdempotentElem_toLinearMap_iff {R M : Type*} [Semiring R] [TopologicalSpace M]
[AddCommMonoid M] [Module R M] {f : M →L[R] M} :
IsIdempotentElem f.toLinearMap ↔ IsIdempotentElem f := by
simp only [IsIdempotentElem, Module.End.mul_eq_comp, ← coe_comp, mul_def, coe_inj]
alias ⟨_, IsIdempotentElem.toLinearMap⟩ := isIdempotentElem_toLinearMap_iff
variable {R M : Type*} [Ring R] [TopologicalSpace M] [AddCommGroup M] [Module R M]
open ContinuousLinearMap
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
isIdempotentElem_toLinearMap_iff
| null |
IsIdempotentElem.ext_iff {p q : M →L[R] M}
(hp : IsIdempotentElem p) (hq : IsIdempotentElem q) :
p = q ↔ range p = range q ∧ ker p = ker q := by
simpa using LinearMap.IsIdempotentElem.ext_iff hp.toLinearMap hq.toLinearMap
alias ⟨_, IsIdempotentElem.ext⟩ := IsIdempotentElem.ext_iff
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
IsIdempotentElem.ext_iff
|
Idempotent operators are equal iff their range and kernels are.
|
IsIdempotentElem.range_mem_invtSubmodule_iff {f T : M →L[R] M}
(hf : IsIdempotentElem f) :
LinearMap.range f ∈ Module.End.invtSubmodule T ↔ f ∘L T ∘L f = T ∘L f := by
simpa [← ContinuousLinearMap.coe_comp] using
LinearMap.IsIdempotentElem.range_mem_invtSubmodule_iff (T := T) hf.toLinearMap
alias ⟨IsIdempotentElem.conj_eq_of_range_mem_invtSubmodule,
IsIdempotentElem.range_mem_invtSubmodule⟩ := IsIdempotentElem.range_mem_invtSubmodule_iff
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
IsIdempotentElem.range_mem_invtSubmodule_iff
|
`range f` is invariant under `T` if and only if `f ∘L T ∘L f = T ∘L f`,
for idempotent `f`.
|
IsIdempotentElem.ker_mem_invtSubmodule_iff {f T : M →L[R] M}
(hf : IsIdempotentElem f) :
LinearMap.ker f ∈ Module.End.invtSubmodule T ↔ f ∘L T ∘L f = f ∘L T := by
simpa [← ContinuousLinearMap.coe_comp] using
LinearMap.IsIdempotentElem.ker_mem_invtSubmodule_iff (T := T) hf.toLinearMap
alias ⟨IsIdempotentElem.conj_eq_of_ker_mem_invtSubmodule,
IsIdempotentElem.ker_mem_invtSubmodule⟩ := IsIdempotentElem.ker_mem_invtSubmodule_iff
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
IsIdempotentElem.ker_mem_invtSubmodule_iff
|
`ker f` is invariant under `T` if and only if `f ∘L T ∘L f = f ∘L T`,
for idempotent `f`.
|
IsIdempotentElem.commute_iff {f T : M →L[R] M}
(hf : IsIdempotentElem f) :
Commute f T ↔ (LinearMap.range f ∈ Module.End.invtSubmodule T
∧ LinearMap.ker f ∈ Module.End.invtSubmodule T) := by
simpa [Commute, SemiconjBy, Module.End.mul_eq_comp, ← coe_comp] using
LinearMap.IsIdempotentElem.commute_iff (T := T) hf.toLinearMap
variable [IsTopologicalAddGroup M]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
IsIdempotentElem.commute_iff
|
An idempotent operator `f` commutes with `T` if and only if
both `range f` and `ker f` are invariant under `T`.
|
IsIdempotentElem.commute_iff_of_isUnit {f T : M →L[R] M} (hT : IsUnit T)
(hf : IsIdempotentElem f) :
Commute f T ↔ (range f).map T = range f ∧ (ker f).map T = ker f := by
have := hT.map ContinuousLinearMap.toLinearMapRingHom
lift T to (M →L[R] M)ˣ using hT
simpa [Commute, SemiconjBy, Module.End.mul_eq_comp, ← ContinuousLinearMap.coe_comp] using
LinearMap.IsIdempotentElem.commute_iff_of_isUnit this hf.toLinearMap
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
IsIdempotentElem.commute_iff_of_isUnit
|
An idempotent operator `f` commutes with an unit operator `T` if and only if
`T (range f) = range f` and `T (ker f) = ker f`.
|
IsIdempotentElem.range_eq_ker {p : M →L[R] M} (hp : IsIdempotentElem p) :
LinearMap.range p = LinearMap.ker (1 - p) :=
LinearMap.IsIdempotentElem.range_eq_ker hp.toLinearMap
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
IsIdempotentElem.range_eq_ker
| null |
IsIdempotentElem.ker_eq_range {p : M →L[R] M} (hp : IsIdempotentElem p) :
LinearMap.ker p = LinearMap.range (1 - p) :=
LinearMap.IsIdempotentElem.ker_eq_range hp.toLinearMap
open ContinuousLinearMap in
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
IsIdempotentElem.ker_eq_range
| null |
IsIdempotentElem.isClosed_range [T1Space M] {p : M →L[R] M}
(hp : IsIdempotentElem p) : IsClosed (LinearMap.range p : Set M) :=
hp.range_eq_ker ▸ isClosed_ker (1 - p)
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
IsIdempotentElem.isClosed_range
| null |
topDualPairing : (E →L[𝕜] 𝕜) →ₗ[𝕜] E →ₗ[𝕜] 𝕜 :=
ContinuousLinearMap.coeLM 𝕜
@[deprecated (since := "2025-08-3")] alias NormedSpace.dualPairing := topDualPairing
@[deprecated (since := "2025-09-03")] alias strongDualPairing := topDualPairing
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
topDualPairing
|
The canonical pairing of a vector space and its topological dual.
|
topDualPairing_apply (v : E →L[𝕜] 𝕜)
(x : E) : topDualPairing 𝕜 E v x = v x :=
rfl
@[deprecated (since := "2025-08-3")] alias NormedSpace.dualPairing_apply := topDualPairing_apply
@[deprecated (since := "2025-09-03")] alias StrongDual.dualPairing_apply := topDualPairing_apply
|
theorem
|
Topology
|
[
"Mathlib.Algebra.Module.LinearMap.DivisionRing",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Algebra.Module.Basic"
] |
Mathlib/Topology/Algebra/Module/LinearMap.lean
|
topDualPairing_apply
| null |
protected prod (f₁ : M₁ →L[R] M₂) (f₂ : M₁ →L[R] M₃) :
M₁ →L[R] M₂ × M₃ :=
⟨(f₁ : M₁ →ₗ[R] M₂).prod f₂, f₁.2.prodMk f₂.2⟩
@[simp, norm_cast]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
prod
|
The Cartesian product of two bounded linear maps, as a bounded linear map.
|
coe_prod (f₁ : M₁ →L[R] M₂) (f₂ : M₁ →L[R] M₃) :
(f₁.prod f₂ : M₁ →ₗ[R] M₂ × M₃) = LinearMap.prod f₁ f₂ :=
rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
coe_prod
| null |
prod_apply (f₁ : M₁ →L[R] M₂) (f₂ : M₁ →L[R] M₃) (x : M₁) :
f₁.prod f₂ x = (f₁ x, f₂ x) :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
prod_apply
| null |
inl : M₁ →L[R] M₁ × M₂ :=
(id R M₁).prod 0
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
inl
|
The left injection into a product is a continuous linear map.
|
inr : M₂ →L[R] M₁ × M₂ :=
(0 : M₂ →L[R] M₁).prod (id R M₂)
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
inr
|
The right injection into a product is a continuous linear map.
|
@[simp]
inl_apply (x : M₁) : inl R M₁ M₂ x = (x, 0) :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
inl_apply
| null |
inr_apply (x : M₂) : inr R M₁ M₂ x = (0, x) :=
rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
inr_apply
| null |
coe_inl : (inl R M₁ M₂ : M₁ →ₗ[R] M₁ × M₂) = LinearMap.inl R M₁ M₂ :=
rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
coe_inl
| null |
coe_inr : (inr R M₁ M₂ : M₂ →ₗ[R] M₁ × M₂) = LinearMap.inr R M₁ M₂ :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
coe_inr
| null |
comp_inl_add_comp_inr (L : M₁ × M₂ →L[R] M₃) (v : M₁ × M₂) :
L.comp (.inl R M₁ M₂) v.1 + L.comp (.inr R M₁ M₂) v.2 = L v := by simp [← map_add]
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
comp_inl_add_comp_inr
| null |
ker_prod (f : M₁ →L[R] M₂) (g : M₁ →L[R] M₃) :
ker (f.prod g) = ker f ⊓ ker g :=
LinearMap.ker_prod (f : M₁ →ₗ[R] M₂) (g : M₁ →ₗ[R] M₃)
variable (R M₁ M₂)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
ker_prod
| null |
fst : M₁ × M₂ →L[R] M₁ where
cont := continuous_fst
toLinearMap := LinearMap.fst R M₁ M₂
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
fst
|
`Prod.fst` as a `ContinuousLinearMap`.
|
snd : M₁ × M₂ →L[R] M₂ where
cont := continuous_snd
toLinearMap := LinearMap.snd R M₁ M₂
variable {R M₁ M₂}
@[simp, norm_cast]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
snd
|
`Prod.snd` as a `ContinuousLinearMap`.
|
coe_fst : ↑(fst R M₁ M₂) = LinearMap.fst R M₁ M₂ :=
rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
coe_fst
| null |
coe_fst' : ⇑(fst R M₁ M₂) = Prod.fst :=
rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
coe_fst'
| null |
coe_snd : ↑(snd R M₁ M₂) = LinearMap.snd R M₁ M₂ :=
rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
coe_snd
| null |
coe_snd' : ⇑(snd R M₁ M₂) = Prod.snd :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
coe_snd'
| null |
fst_prod_snd : (fst R M₁ M₂).prod (snd R M₁ M₂) = id R (M₁ × M₂) :=
ext fun ⟨_x, _y⟩ => rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
fst_prod_snd
| null |
fst_comp_prod (f : M₁ →L[R] M₂) (g : M₁ →L[R] M₃) :
(fst R M₂ M₃).comp (f.prod g) = f :=
ext fun _x => rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
fst_comp_prod
| null |
snd_comp_prod (f : M₁ →L[R] M₂) (g : M₁ →L[R] M₃) :
(snd R M₂ M₃).comp (f.prod g) = g :=
ext fun _x => rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
snd_comp_prod
| null |
prodMap (f₁ : M₁ →L[R] M₂) (f₂ : M₃ →L[R] M₄) :
M₁ × M₃ →L[R] M₂ × M₄ :=
(f₁.comp (fst R M₁ M₃)).prod (f₂.comp (snd R M₁ M₃))
@[simp, norm_cast]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
prodMap
|
`Prod.map` of two continuous linear maps.
|
coe_prodMap (f₁ : M₁ →L[R] M₂)
(f₂ : M₃ →L[R] M₄) : ↑(f₁.prodMap f₂) = (f₁ : M₁ →ₗ[R] M₂).prodMap (f₂ : M₃ →ₗ[R] M₄) :=
rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
coe_prodMap
| null |
coe_prodMap' (f₁ : M₁ →L[R] M₂)
(f₂ : M₃ →L[R] M₄) : ⇑(f₁.prodMap f₂) = Prod.map f₁ f₂ :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
coe_prodMap'
| null |
pi (f : ∀ i, M →L[R] φ i) : M →L[R] ∀ i, φ i :=
⟨LinearMap.pi fun i => f i, continuous_pi fun i => (f i).continuous⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
pi
|
`pi` construction for continuous linear functions. From a family of continuous linear functions
it produces a continuous linear function into a family of topological modules.
|
coe_pi' (f : ∀ i, M →L[R] φ i) : ⇑(pi f) = fun c i => f i c :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
coe_pi'
| null |
coe_pi (f : ∀ i, M →L[R] φ i) : (pi f : M →ₗ[R] ∀ i, φ i) = LinearMap.pi fun i => f i :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
coe_pi
| null |
pi_apply (f : ∀ i, M →L[R] φ i) (c : M) (i : ι) : pi f c i = f i c :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
pi_apply
| null |
pi_eq_zero (f : ∀ i, M →L[R] φ i) : pi f = 0 ↔ ∀ i, f i = 0 := by
simp only [ContinuousLinearMap.ext_iff, pi_apply, funext_iff]
exact forall_swap
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
pi_eq_zero
| null |
pi_zero : pi (fun _ => 0 : ∀ i, M →L[R] φ i) = 0 :=
ext fun _ => rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
pi_zero
| null |
pi_comp (f : ∀ i, M →L[R] φ i) (g : M₂ →L[R] M) :
(pi f).comp g = pi fun i => (f i).comp g :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
pi_comp
| null |
proj (i : ι) : (∀ i, φ i) →L[R] φ i :=
⟨LinearMap.proj i, continuous_apply _⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
proj
|
The projections from a family of topological modules are continuous linear maps.
|
proj_apply (i : ι) (b : ∀ i, φ i) : (proj i : (∀ i, φ i) →L[R] φ i) b = b i :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
proj_apply
| null |
proj_pi (f : ∀ i, M₂ →L[R] φ i) (i : ι) : (proj i).comp (pi f) = f i := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
proj_pi
| null |
coe_proj (i : ι) : (proj i).toLinearMap = (LinearMap.proj i : ((i : ι) → φ i) →ₗ[R] _) :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
coe_proj
| null |
pi_proj : pi proj = .id R (∀ i, φ i) := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
pi_proj
| null |
pi_proj_comp (f : M₂ →L[R] ∀ i, φ i) : pi (proj · ∘L f) = f := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
pi_proj_comp
| null |
iInf_ker_proj : (⨅ i, ker (proj i : (∀ i, φ i) →L[R] φ i) : Submodule R (∀ i, φ i)) = ⊥ :=
LinearMap.iInf_ker_proj
variable (R φ)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
iInf_ker_proj
| null |
_root_.Pi.compRightL {α : Type*} (f : α → ι) : ((i : ι) → φ i) →L[R] ((i : α) → φ (f i)) where
toFun := fun v i ↦ v (f i)
map_add' := by intros; ext; simp
map_smul' := by intros; ext; simp
cont := by fun_prop
@[simp] lemma _root_.Pi.compRightL_apply {α : Type*} (f : α → ι) (v : (i : ι) → φ i) (i : α) :
Pi.compRightL R φ f v i = v (f i) := rfl
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
_root_.Pi.compRightL
|
Given a function `f : α → ι`, it induces a continuous linear function by right composition on
product types. For `f = Subtype.val`, this corresponds to forgetting some set of variables.
|
@[simps! -fullyApplied]
single [DecidableEq ι] (i : ι) : φ i →L[R] (∀ i, φ i) where
toLinearMap := .single R φ i
cont := continuous_single _
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
single
|
`Pi.single` as a bundled continuous linear map.
|
sum_comp_single [Fintype ι] [DecidableEq ι] (L : (Π i, φ i) →L[R] M) (v : Π i, φ i) :
∑ i, L.comp (.single R φ i) (v i) = L v := by
simp [← map_sum, LinearMap.sum_single_apply]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
sum_comp_single
| null |
range_prod_eq {f : M →L[R] M₂} {g : M →L[R] M₃} (h : ker f ⊔ ker g = ⊤) :
range (f.prod g) = (range f).prod (range g) :=
LinearMap.range_prod_eq h
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
range_prod_eq
| null |
ker_prod_ker_le_ker_coprod (f : M →L[R] M₃) (g : M₂ →L[R] M₃) :
(LinearMap.ker f).prod (LinearMap.ker g) ≤ LinearMap.ker (f.coprod g) :=
LinearMap.ker_prod_ker_le_ker_coprod f.toLinearMap g.toLinearMap
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
ker_prod_ker_le_ker_coprod
| null |
@[simps apply]
prodEquiv : (M →L[R] M₂) × (M →L[R] M₃) ≃ (M →L[R] M₂ × M₃) where
toFun f := f.1.prod f.2
invFun f := ⟨(fst _ _ _).comp f, (snd _ _ _).comp f⟩
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
prodEquiv
|
`ContinuousLinearMap.prod` as an `Equiv`.
|
prod_ext_iff {f g : M × M₂ →L[R] M₃} :
f = g ↔ f.comp (inl _ _ _) = g.comp (inl _ _ _) ∧ f.comp (inr _ _ _) = g.comp (inr _ _ _) := by
simp only [← coe_inj, LinearMap.prod_ext_iff]
rfl
@[ext]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean
|
prod_ext_iff
| null |
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