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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