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 ⌀ |
|---|---|---|---|---|---|---|
prod_ext {f g : M × M₂ →L[R] M₃} (hl : f.comp (inl _ _ _) = g.comp (inl _ _ _))
(hr : f.comp (inr _ _ _) = g.comp (inr _ _ _)) : f = g :=
prod_ext_iff.2 ⟨hl, hr⟩
variable (S : Type*) [Semiring S]
[Module S M₂] [ContinuousAdd M₂] [SMulCommClass R S M₂] [ContinuousConstSMul S M₂]
[Module S M₃] [ContinuousAdd M₃... | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean | prod_ext | null |
@[simps apply]
prodₗ : ((M →L[R] M₂) × (M →L[R] M₃)) ≃ₗ[S] M →L[R] M₂ × M₃ :=
{ prodEquiv with
map_add' := fun _f _g => rfl
map_smul' := fun _c _f => rfl } | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean | prodₗ | `ContinuousLinearMap.prod` as a `LinearEquiv`. |
@[simps! coe apply]
coprod (f₁ : M₁ →L[R] M) (f₂ : M₂ →L[R] M) : M₁ × M₂ →L[R] M :=
⟨.coprod f₁ f₂, (f₁.cont.comp continuous_fst).add (f₂.cont.comp continuous_snd)⟩
@[simp] lemma coprod_add (f₁ g₁ : M₁ →L[R] M) (f₂ g₂ : M₂ →L[R] M) :
(f₁ + g₁).coprod (f₂ + g₂) = f₁.coprod f₂ + g₁.coprod g₂ := by ext <;> simp | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean | coprod | The continuous linear map given by `(x, y) ↦ f₁ x + f₂ y`. |
range_coprod (f₁ : M₁ →L[R] M) (f₂ : M₂ →L[R] M) :
range (f₁.coprod f₂) = range f₁ ⊔ range f₂ := LinearMap.range_coprod .. | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean | range_coprod | null |
comp_fst_add_comp_snd (f₁ : M₁ →L[R] M) (f₂ : M₂ →L[R] M) :
f₁.comp (.fst _ _ _) + f₂.comp (.snd _ _ _) = f₁.coprod f₂ := rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean | comp_fst_add_comp_snd | null |
comp_coprod (f : M →L[R] N) (g₁ : M₁ →L[R] M) (g₂ : M₂ →L[R] M) :
f.comp (g₁.coprod g₂) = (f.comp g₁).coprod (f.comp g₂) :=
coe_injective <| LinearMap.comp_coprod ..
@[simp] lemma coprod_comp_inl (f₁ : M₁ →L[R] M) (f₂ : M₂ →L[R] M) :
(f₁.coprod f₂).comp (.inl _ _ _) = f₁ := coe_injective <| LinearMap.coprod_i... | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean | comp_coprod | null |
coprod_inl_inr : ContinuousLinearMap.coprod (.inl R M N) (.inr R M N) = .id R (M × N) :=
coe_injective <| LinearMap.coprod_inl_inr | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean | coprod_inl_inr | null |
@[simps]
coprodEquiv [ContinuousAdd M₁] [ContinuousAdd M₂] [Semiring S] [Module S M]
[ContinuousConstSMul S M] [SMulCommClass R S M] :
((M₁ →L[R] M) × (M₂ →L[R] M)) ≃ₗ[S] M₁ × M₂ →L[R] M where
toFun f := f.1.coprod f.2
invFun f := (f.comp (.inl ..), f.comp (.inr ..))
left_inv f := by simp
right_inv f :=... | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean | coprodEquiv | Taking the product of two maps with the same codomain is equivalent to taking the product of
their domains.
See note [bundled maps over different rings] for why separate `R` and `S` semirings are used.
TODO: Upgrade this to a `ContinuousLinearEquiv`. This should be true for any topological
vector space over a normed f... |
ker_coprod_of_disjoint_range {f₁ : M₁ →L[R] M} {f₂ : M₂ →L[R] M}
(hf : Disjoint (range f₁) (range f₂)) :
LinearMap.ker (f₁.coprod f₂) = (LinearMap.ker f₁).prod (LinearMap.ker f₂) :=
LinearMap.ker_coprod_of_disjoint_range f₁.toLinearMap f₂.toLinearMap hf | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/LinearMapPiProd.lean | ker_coprod_of_disjoint_range | null |
IsClosed (f : E →ₗ.[R] F) : Prop :=
_root_.IsClosed (f.graph : Set (E × F))
variable [ContinuousAdd E] [ContinuousAdd F]
variable [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] | def | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | IsClosed | An unbounded operator is closed iff its graph is closed. |
IsClosable (f : E →ₗ.[R] F) : Prop :=
∃ f' : LinearPMap R E F, f.graph.topologicalClosure = f'.graph | def | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | IsClosable | An unbounded operator is closable iff the closure of its graph is a graph. |
IsClosed.isClosable {f : E →ₗ.[R] F} (hf : f.IsClosed) : f.IsClosable :=
⟨f, hf.submodule_topologicalClosure_eq⟩ | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | IsClosed.isClosable | A closed operator is trivially closable. |
IsClosable.leIsClosable {f g : E →ₗ.[R] F} (hf : f.IsClosable) (hfg : g ≤ f) :
g.IsClosable := by
obtain ⟨f', hf⟩ := hf
have : g.graph.topologicalClosure ≤ f'.graph := by
rw [← hf]
exact Submodule.topologicalClosure_mono (le_graph_of_le hfg)
use g.graph.topologicalClosure.toLinearPMap
rw [Submodule.... | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | IsClosable.leIsClosable | If `g` has a closable extension `f`, then `g` itself is closable. |
IsClosable.existsUnique {f : E →ₗ.[R] F} (hf : f.IsClosable) :
∃! f' : E →ₗ.[R] F, f.graph.topologicalClosure = f'.graph := by
refine existsUnique_of_exists_of_unique hf fun _ _ hy₁ hy₂ => eq_of_eq_graph ?_
rw [← hy₁, ← hy₂]
open Classical in | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | IsClosable.existsUnique | The closure is unique. |
noncomputable closure (f : E →ₗ.[R] F) : E →ₗ.[R] F :=
if hf : f.IsClosable then hf.choose else f | def | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | closure | If `f` is closable, then `f.closure` is the closure. Otherwise it is defined
as `f.closure = f`. |
closure_def {f : E →ₗ.[R] F} (hf : f.IsClosable) : f.closure = hf.choose := by
simp [closure, hf] | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | closure_def | null |
closure_def' {f : E →ₗ.[R] F} (hf : ¬f.IsClosable) : f.closure = f := by simp [closure, hf] | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | closure_def' | null |
IsClosable.graph_closure_eq_closure_graph {f : E →ₗ.[R] F} (hf : f.IsClosable) :
f.graph.topologicalClosure = f.closure.graph := by
rw [closure_def hf]
exact hf.choose_spec | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | IsClosable.graph_closure_eq_closure_graph | The closure (as a submodule) of the graph is equal to the graph of the closure
(as a `LinearPMap`). |
le_closure (f : E →ₗ.[R] F) : f ≤ f.closure := by
by_cases hf : f.IsClosable
· refine le_of_le_graph ?_
rw [← hf.graph_closure_eq_closure_graph]
exact (graph f).le_topologicalClosure
rw [closure_def' hf] | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | le_closure | A `LinearPMap` is contained in its closure. |
IsClosable.closure_mono {f g : E →ₗ.[R] F} (hg : g.IsClosable) (h : f ≤ g) :
f.closure ≤ g.closure := by
refine le_of_le_graph ?_
rw [← (hg.leIsClosable h).graph_closure_eq_closure_graph]
rw [← hg.graph_closure_eq_closure_graph]
exact Submodule.topologicalClosure_mono (le_graph_of_le h) | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | IsClosable.closure_mono | null |
IsClosable.closure_isClosed {f : E →ₗ.[R] F} (hf : f.IsClosable) : f.closure.IsClosed := by
rw [IsClosed, ← hf.graph_closure_eq_closure_graph]
exact f.graph.isClosed_topologicalClosure | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | IsClosable.closure_isClosed | If `f` is closable, then the closure is closed. |
IsClosable.closureIsClosable {f : E →ₗ.[R] F} (hf : f.IsClosable) : f.closure.IsClosable :=
hf.closure_isClosed.isClosable | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | IsClosable.closureIsClosable | If `f` is closable, then the closure is closable. |
isClosable_iff_exists_closed_extension {f : E →ₗ.[R] F} :
f.IsClosable ↔ ∃ g : E →ₗ.[R] F, g.IsClosed ∧ f ≤ g :=
⟨fun h => ⟨f.closure, h.closure_isClosed, f.le_closure⟩, fun ⟨_, hg, h⟩ =>
hg.isClosable.leIsClosable h⟩
/-! ### The core of a linear operator -/ | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | isClosable_iff_exists_closed_extension | null |
HasCore (f : E →ₗ.[R] F) (S : Submodule R E) : Prop where
le_domain : S ≤ f.domain
closure_eq : (f.domRestrict S).closure = f | structure | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | HasCore | A submodule `S` is a core of `f` if the closure of the restriction of `f` to `S` is `f`. |
hasCore_def {f : E →ₗ.[R] F} {S : Submodule R E} (h : f.HasCore S) :
(f.domRestrict S).closure = f :=
h.2 | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | hasCore_def | null |
closureHasCore (f : E →ₗ.[R] F) : f.closure.HasCore f.domain := by
refine ⟨f.le_closure.1, ?_⟩
congr
ext x h1 h2
· simp only [domRestrict_domain, Submodule.mem_inf, and_iff_left_iff_imp]
intro hx
exact f.le_closure.1 hx
let z : f.closure.domain := ⟨x, f.le_closure.1 h2⟩
have hyz : x = z := rfl
rw ... | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | closureHasCore | For every unbounded operator `f` the submodule `f.domain` is a core of its closure.
Note that we don't require that `f` is closable, due to the definition of the closure. |
inverse_closed_iff (hf : LinearMap.ker f.toFun = ⊥) : f.inverse.IsClosed ↔ f.IsClosed := by
rw [IsClosed, inverse_graph hf]
exact (ContinuousLinearEquiv.prodComm R E F).isClosed_image
variable [ContinuousAdd E] [ContinuousAdd F]
variable [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | inverse_closed_iff | The inverse of `f : LinearPMap` is closed if and only if `f` is closed. |
closure_inverse_graph (hf : LinearMap.ker f.toFun = ⊥) (hf' : f.IsClosable)
(hcf : LinearMap.ker f.closure.toFun = ⊥) :
f.closure.inverse.graph = f.inverse.graph.topologicalClosure := by
rw [inverse_graph hf, inverse_graph hcf, ← hf'.graph_closure_eq_closure_graph]
apply SetLike.ext'
simp only [Submodule.... | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | closure_inverse_graph | If `f` is invertible and closable as well as its closure being invertible, then
the graph of the inverse of the closure is given by the closure of the graph of the inverse. |
inverse_isClosable_iff (hf : LinearMap.ker f.toFun = ⊥) (hf' : f.IsClosable) :
f.inverse.IsClosable ↔ LinearMap.ker f.closure.toFun = ⊥ := by
constructor
· intro ⟨f', h⟩
rw [LinearMap.ker_eq_bot']
intro ⟨x, hx⟩ hx'
simp only [Submodule.mk_eq_zero]
rw [toFun_eq_coe, eq_comm, image_iff] at hx'
... | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | inverse_isClosable_iff | Assuming that `f` is invertible and closable, then the closure is invertible if and only
if the inverse of `f` is closable. |
inverse_closure (hf : LinearMap.ker f.toFun = ⊥) (hf' : f.IsClosable)
(hcf : LinearMap.ker f.closure.toFun = ⊥) :
f.inverse.closure = f.closure.inverse := by
apply eq_of_eq_graph
rw [closure_inverse_graph hf hf' hcf,
((inverse_isClosable_iff hf hf').mpr hcf).graph_closure_eq_closure_graph] | theorem | Topology | [
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Equiv"
] | Mathlib/Topology/Algebra/Module/LinearPMap.lean | inverse_closure | If `f` is invertible and closable, then taking the closure and the inverse commute. |
LocallyConvexSpace (𝕜 E : Type*) [Semiring 𝕜] [PartialOrder 𝕜]
[AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : Prop where
convex_basis : ∀ x : E, (𝓝 x).HasBasis (fun s : Set E => s ∈ 𝓝 x ∧ Convex 𝕜 s) id
variable (𝕜 E : Type*) [Semiring 𝕜] [PartialOrder 𝕜]
[AddCommMonoid E] [Module 𝕜 E] [Topolo... | class | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | LocallyConvexSpace | A `LocallyConvexSpace` is a topological semimodule over an ordered semiring in which convex
neighborhoods of a point form a neighborhood basis at that point. |
locallyConvexSpace_iff :
LocallyConvexSpace 𝕜 E ↔ ∀ x : E, (𝓝 x).HasBasis (fun s : Set E => s ∈ 𝓝 x ∧ Convex 𝕜 s) id :=
⟨fun _ ↦ LocallyConvexSpace.convex_basis, LocallyConvexSpace.mk⟩ | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | locallyConvexSpace_iff | null |
LocallyConvexSpace.ofBases {ι : Type*} (b : E → ι → Set E) (p : E → ι → Prop)
(hbasis : ∀ x : E, (𝓝 x).HasBasis (p x) (b x)) (hconvex : ∀ x i, p x i → Convex 𝕜 (b x i)) :
LocallyConvexSpace 𝕜 E :=
⟨fun x =>
(hbasis x).to_hasBasis
(fun i hi => ⟨b x i, ⟨⟨(hbasis x).mem_of_mem hi, hconvex x i hi⟩, l... | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | LocallyConvexSpace.ofBases | null |
LocallyConvexSpace.convex_basis_zero [LocallyConvexSpace 𝕜 E] :
(𝓝 0 : Filter E).HasBasis (fun s => s ∈ (𝓝 0 : Filter E) ∧ Convex 𝕜 s) id :=
LocallyConvexSpace.convex_basis 0 | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | LocallyConvexSpace.convex_basis_zero | null |
locallyConvexSpace_iff_exists_convex_subset :
LocallyConvexSpace 𝕜 E ↔ ∀ x : E, ∀ U ∈ 𝓝 x, ∃ S ∈ 𝓝 x, Convex 𝕜 S ∧ S ⊆ U :=
(locallyConvexSpace_iff 𝕜 E).trans (forall_congr' fun _ => hasBasis_self) | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | locallyConvexSpace_iff_exists_convex_subset | null |
LocallyConvexSpace.ofBasisZero {ι : Type*} (b : ι → Set E) (p : ι → Prop)
(hbasis : (𝓝 0).HasBasis p b) (hconvex : ∀ i, p i → Convex 𝕜 (b i)) :
LocallyConvexSpace 𝕜 E := by
refine LocallyConvexSpace.ofBases 𝕜 E (fun (x : E) (i : ι) => (x + ·) '' b i) (fun _ => p)
(fun x => ?_) fun x i hi => (hconvex i... | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | LocallyConvexSpace.ofBasisZero | null |
locallyConvexSpace_iff_zero : LocallyConvexSpace 𝕜 E ↔
(𝓝 0 : Filter E).HasBasis (fun s : Set E => s ∈ (𝓝 0 : Filter E) ∧ Convex 𝕜 s) id :=
⟨fun _ => LocallyConvexSpace.convex_basis 0, fun h =>
LocallyConvexSpace.ofBasisZero 𝕜 E _ _ h fun _ => And.right⟩ | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | locallyConvexSpace_iff_zero | null |
locallyConvexSpace_iff_exists_convex_subset_zero :
LocallyConvexSpace 𝕜 E ↔ ∀ U ∈ (𝓝 0 : Filter E), ∃ S ∈ (𝓝 0 : Filter E), Convex 𝕜 S ∧ S ⊆ U :=
(locallyConvexSpace_iff_zero 𝕜 E).trans hasBasis_self | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | locallyConvexSpace_iff_exists_convex_subset_zero | null |
Convex.locPathConnectedSpace [Module ℝ E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E]
{S : Set E} (hS : Convex ℝ S) : LocPathConnectedSpace S := by
refine ⟨fun x ↦ ⟨fun s ↦ ⟨fun hs ↦ ?_, fun ⟨t, ht⟩ ↦ mem_of_superset ht.1.1 ht.2⟩⟩⟩
let ⟨t, ht⟩ := (mem_nhds_subtype S x s).mp hs
let ⟨t', ht'⟩ := (LocallyConve... | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | Convex.locPathConnectedSpace | Convex subsets of locally convex spaces are locally path-connected. |
LocallyConvexSpace.convex_open_basis_zero [LocallyConvexSpace 𝕜 E] :
(𝓝 0 : Filter E).HasBasis (fun s => (0 : E) ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) id :=
(LocallyConvexSpace.convex_basis_zero 𝕜 E).to_hasBasis
(fun s hs =>
⟨interior s, ⟨mem_interior_iff_mem_nhds.mpr hs.1, isOpen_interior, hs.2.interior⟩,
... | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | LocallyConvexSpace.convex_open_basis_zero | null |
Disjoint.exists_open_convexes (disj : Disjoint s t)
(hs₁ : Convex 𝕜 s) (hs₂ : IsCompact s) (ht₁ : Convex 𝕜 t) (ht₂ : IsClosed t) :
∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v := by
letI : UniformSpace E := IsTopologicalAddGroup.toUniformSpace E
haveI : IsUniformAd... | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | Disjoint.exists_open_convexes | In a locally convex space, every two disjoint convex sets such that one is compact and the other
is closed admit disjoint convex open neighborhoods. |
exists_open_convex_of_notMem (hx : x ∉ s) (hsconv : Convex 𝕜 s) (hsclosed : IsClosed s) :
∃ U V : Set E,
IsOpen U ∧ IsOpen V ∧ Convex 𝕜 U ∧ Convex 𝕜 V ∧ x ∈ U ∧ s ⊆ V ∧ Disjoint U V := by
simpa [*] using Disjoint.exists_open_convexes (s := {x}) (t := s) (𝕜 := 𝕜)
@[deprecated (since := "2025-05-23")]
al... | lemma | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | exists_open_convex_of_notMem | In a locally convex space, every point `x` and closed convex set `s ∌ x` admit disjoint convex
open neighborhoods. |
protected LocallyConvexSpace.sInf {ts : Set (TopologicalSpace E)}
(h : ∀ t ∈ ts, @LocallyConvexSpace 𝕜 E _ _ _ _ t) :
@LocallyConvexSpace 𝕜 E _ _ _ _ (sInf ts) := by
letI : TopologicalSpace E := sInf ts
refine .ofBases 𝕜 E (fun _ => fun If : Set ts × (ts → Set E) => ⋂ i ∈ If.1, If.2 i)
(fun x => fu... | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | LocallyConvexSpace.sInf | null |
protected LocallyConvexSpace.iInf {ts' : ι → TopologicalSpace E}
(h' : ∀ i, @LocallyConvexSpace 𝕜 E _ _ _ _ (ts' i)) :
@LocallyConvexSpace 𝕜 E _ _ _ _ (⨅ i, ts' i) :=
.sInf <| by rwa [forall_mem_range]
@[deprecated (since := "2025-05-05")]
alias locallyConvexSpace_iInf := LocallyConvexSpace.iInf | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | LocallyConvexSpace.iInf | null |
protected LocallyConvexSpace.inf {t₁ t₂ : TopologicalSpace E}
(h₁ : @LocallyConvexSpace 𝕜 E _ _ _ _ t₁)
(h₂ : @LocallyConvexSpace 𝕜 E _ _ _ _ t₂) : @LocallyConvexSpace 𝕜 E _ _ _ _ (t₁ ⊓ t₂) := by
rw [inf_eq_iInf]
refine .iInf fun b => ?_
cases b <;> assumption
@[deprecated (since := "2025-05-05")]
alia... | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | LocallyConvexSpace.inf | null |
protected LocallyConvexSpace.induced {t : TopologicalSpace F} [LocallyConvexSpace 𝕜 F]
(f : E →ₗ[𝕜] F) : @LocallyConvexSpace 𝕜 E _ _ _ _ (t.induced f) := by
letI : TopologicalSpace E := t.induced f
refine LocallyConvexSpace.ofBases 𝕜 E (fun _ => preimage f)
(fun x => fun s : Set F => s ∈ 𝓝 (f x) ∧ Conv... | theorem | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | LocallyConvexSpace.induced | null |
Pi.locallyConvexSpace {ι : Type*} {X : ι → Type*} [∀ i, AddCommMonoid (X i)]
[∀ i, TopologicalSpace (X i)] [∀ i, Module 𝕜 (X i)] [∀ i, LocallyConvexSpace 𝕜 (X i)] :
LocallyConvexSpace 𝕜 (∀ i, X i) :=
.iInf fun i => .induced (LinearMap.proj i) | instance | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | Pi.locallyConvexSpace | null |
Prod.locallyConvexSpace [TopologicalSpace E] [TopologicalSpace F] [LocallyConvexSpace 𝕜 E]
[LocallyConvexSpace 𝕜 F] : LocallyConvexSpace 𝕜 (E × F) :=
.inf (.induced (LinearMap.fst _ _ _)) (.induced (LinearMap.snd _ _ _)) | instance | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | Prod.locallyConvexSpace | null |
LinearOrderedSemiring.toLocallyConvexSpace {R : Type*} [TopologicalSpace R]
[Semiring R] [LinearOrder R] [IsStrictOrderedRing R] [OrderTopology R] :
LocallyConvexSpace R R where
convex_basis x := by
obtain hl | hl := isBot_or_exists_lt x
· refine hl.rec ?_ _
intro
refine nhds_bot_basis.to_... | instance | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | LinearOrderedSemiring.toLocallyConvexSpace | A linear ordered semiring is a locally convex space over itself. |
Convex.eventually_nhdsWithin_segment {E 𝕜 : Type*}
[Semiring 𝕜] [PartialOrder 𝕜]
[AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [LocallyConvexSpace 𝕜 E]
{s : Set E} (hs : Convex 𝕜 s) {x₀ : E} (hx₀s : x₀ ∈ s)
{p : E → Prop} (h : ∀ᶠ x in 𝓝[s] x₀, p x) :
∀ᶠ x in 𝓝[s] x₀, ∀ y ∈ segment 𝕜 x... | lemma | Topology | [
"Mathlib.Analysis.Convex.Topology",
"Mathlib.Topology.Connected.LocPathConnected",
"Mathlib.Analysis.Convex.PathConnected"
] | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | Convex.eventually_nhdsWithin_segment | null |
moduleTopology : TopologicalSpace A :=
sInf {t | @ContinuousSMul R A _ _ t ∧ @ContinuousAdd A t _} | abbrev | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | moduleTopology | The module topology, for a module `A` over a topological ring `R`. It's the finest topology
making addition and the `R`-action continuous, or equivalently the finest topology making `A`
into a topological `R`-module. More precisely it's the Inf of the set of
topologies with these properties; theorems `continuousSMul` a... |
IsModuleTopology [τA : TopologicalSpace A] : Prop where
/-- Note that this should not be used directly, and `eq_moduleTopology`, which takes `R` and `A`
explicitly, should be used instead. -/
eq_moduleTopology' : τA = moduleTopology R A | class | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | IsModuleTopology | A class asserting that the topology on a module over a topological ring `R` is
the module topology. See `moduleTopology` for more discussion of the module topology. |
eq_moduleTopology [τA : TopologicalSpace A] [IsModuleTopology R A] :
τA = moduleTopology R A :=
IsModuleTopology.eq_moduleTopology' (R := R) (A := A) | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | eq_moduleTopology | null |
ModuleTopology.continuousSMul : @ContinuousSMul R A _ _ (moduleTopology R A) :=
/- Proof: We need to prove that the product topology is finer than the pullback
of the module topology. But the module topology is an Inf and thus a limit,
and pullback is a right adjoint, so it preserves limits.
We must th... | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | ModuleTopology.continuousSMul | Note that the topology isn't part of the discrimination key so this gets tried on every
`IsModuleTopology` goal and hence the low priority.
-/
instance (priority := low) {R : Type*} [TopologicalSpace R] {A : Type*} [Add A] [SMul R A] :
letI := moduleTopology R A; IsModuleTopology R A :=
letI := moduleTopology R A... |
ModuleTopology.continuousAdd : @ContinuousAdd A (moduleTopology R A) _ :=
continuousAdd_sInf fun _ h ↦ h.2 | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | ModuleTopology.continuousAdd | Addition `+ : A × A → A` is continuous if `R` is a topological
ring, and `A` is an `R` module with the module topology. |
IsModuleTopology.toContinuousSMul [TopologicalSpace A] [IsModuleTopology R A] :
ContinuousSMul R A := eq_moduleTopology R A ▸ ModuleTopology.continuousSMul R A | instance | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | IsModuleTopology.toContinuousSMul | null |
IsModuleTopology.toContinuousAdd [TopologicalSpace A] [IsModuleTopology R A] :
ContinuousAdd A := eq_moduleTopology R A ▸ ModuleTopology.continuousAdd R A | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | IsModuleTopology.toContinuousAdd | null |
moduleTopology_le [τA : TopologicalSpace A] [ContinuousSMul R A] [ContinuousAdd A] :
moduleTopology R A ≤ τA := sInf_le ⟨inferInstance, inferInstance⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | moduleTopology_le | The module topology is `≤` any topology making `A` into a topological module. |
of_continuous_id [ContinuousAdd A] [ContinuousSMul R A]
(h : @Continuous A A τA (moduleTopology R A) id) :
IsModuleTopology R A where
eq_moduleTopology' := le_antisymm (continuous_id_iff_le.1 h) (moduleTopology_le _ _) | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | of_continuous_id | If `A` is a topological `R`-module and the identity map from (`A` with its given
topology) to (`A` with the module topology) is continuous, then the topology on `A` is
the module topology. |
instSubsingleton [Subsingleton A] : IsModuleTopology R A where
eq_moduleTopology' := Subsingleton.elim _ _ | instance | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | instSubsingleton | The zero module has the module topology. |
iso (e : A ≃L[R] B) : IsModuleTopology R B where
eq_moduleTopology' := by
let g : A →ₗ[R] B := e
let g' : B →ₗ[R] A := e.symm
let h : A →+ B := e
let h' : B →+ A := e.symm
simp_rw [e.toHomeomorph.symm.isInducing.1, eq_moduleTopology R A, moduleTopology, induced_sInf]
apply congr_arg
ext τ ... | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | iso | If `A` and `B` are `R`-modules, homeomorphic via an `R`-linear homeomorphism, and if
`A` has the module topology, then so does `B`. |
_root_.IsTopologicalSemiring.toIsModuleTopology : IsModuleTopology R R := by
/- By a previous lemma it suffices to show that the identity from (R,usual) to
(R, module topology) is continuous. -/
apply of_continuous_id
/-
The idea needed here is to rewrite the identity function as the composite of `r ↦ (r,1)`
... | instance | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | _root_.IsTopologicalSemiring.toIsModuleTopology | The topology on a topological semiring `R` agrees with the module topology when considering
`R` as an `R`-module in the obvious way (i.e., via `Semiring.toModule`). |
_root_.IsTopologicalSemiring.toOppositeIsModuleTopology : IsModuleTopology Rᵐᵒᵖ R :=
.iso (MulOpposite.opContinuousLinearEquiv Rᵐᵒᵖ).symm | instance | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | _root_.IsTopologicalSemiring.toOppositeIsModuleTopology | The module topology coming from the action of the topological ring `Rᵐᵒᵖ` on `R`
(via `Semiring.toOppositeModule`, i.e. via `(op r) • m = m * r`) is `R`'s topology. |
@[fun_prop, continuity]
continuous_of_distribMulActionHom (φ : A →+[R] B) : Continuous φ := by
rw [eq_moduleTopology R A, continuous_iff_le_induced]
exact sInf_le <| ⟨continuousSMul_induced (φ.toMulActionHom),
continuousAdd_induced φ.toAddMonoidHom⟩
@[fun_prop, continuity] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | continuous_of_distribMulActionHom | Every `R`-linear map between two topological `R`-modules, where the source has the module
topology, is continuous. |
continuous_of_linearMap (φ : A →ₗ[R] B) : Continuous φ :=
continuous_of_distribMulActionHom φ.toDistribMulActionHom
variable (R) in | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | continuous_of_linearMap | null |
continuous_neg (C : Type*) [AddCommGroup C] [Module R C] [TopologicalSpace C]
[IsModuleTopology R C] : Continuous (fun a ↦ -a : C → C) :=
haveI : ContinuousAdd C := IsModuleTopology.toContinuousAdd R C
continuous_of_linearMap (LinearEquiv.neg R).toLinearMap
variable (R) in | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | continuous_neg | null |
continuousNeg (C : Type*) [AddCommGroup C] [Module R C] [TopologicalSpace C]
[IsModuleTopology R C] : ContinuousNeg C where
continuous_neg := continuous_neg R C
variable (R) in | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | continuousNeg | null |
topologicalAddGroup (C : Type*) [AddCommGroup C] [Module R C] [TopologicalSpace C]
[IsModuleTopology R C] : IsTopologicalAddGroup C where
continuous_add := (IsModuleTopology.toContinuousAdd R C).1
continuous_neg := continuous_neg R C
@[fun_prop, continuity] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | topologicalAddGroup | null |
continuous_of_ringHom {R A B} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B]
[TopologicalSpace R] [TopologicalSpace A] [IsModuleTopology R A] [TopologicalSpace B]
[IsTopologicalSemiring B]
(φ : A →+* B) (hφ : Continuous (φ.comp (algebraMap R A))) : Continuous φ := by
let inst := Module.compHom B... | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | continuous_of_ringHom | null |
isQuotientMap_of_surjective [τB : TopologicalSpace B] [IsModuleTopology R B]
{φ : A →ₗ[R] B} (hφ : Function.Surjective φ) :
IsQuotientMap φ where
surjective := hφ
eq_coinduced := by
haveI := topologicalAddGroup R A
haveI := topologicalAddGroup R B
have this : Continuous φ := continuous_of_linear... | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | isQuotientMap_of_surjective | A linear surjection between modules with the module topology is a quotient map.
Equivalently, the pushforward of the module topology along a surjective linear map is
again the module topology. |
isOpenQuotientMap_of_surjective [TopologicalSpace B] [IsModuleTopology R B]
{φ : A →ₗ[R] B} (hφ : Function.Surjective φ) :
IsOpenQuotientMap φ :=
have := toContinuousAdd R A
AddMonoidHom.isOpenQuotientMap_of_isQuotientMap <| isQuotientMap_of_surjective hφ
omit [IsModuleTopology R A] in | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | isOpenQuotientMap_of_surjective | A linear surjection between modules with the module topology is an open quotient map. |
isOpenMap_of_surjective [TopologicalSpace B] [IsModuleTopology R B]
[ContinuousAdd A] [ContinuousSMul R A] {φ : A →ₗ[R] B} (hφ : Function.Surjective φ) :
IsOpenMap φ := by
have hOpenMap :=
letI : TopologicalSpace A := moduleTopology R A
have : IsModuleTopology R A := ⟨rfl⟩
isOpenQuotientMap_of_sur... | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | isOpenMap_of_surjective | A linear surjection to a module with the module topology is open. |
_root_.ModuleTopology.eq_coinduced_of_surjective
{φ : A →ₗ[R] B} (hφ : Function.Surjective φ) :
moduleTopology R B = TopologicalSpace.coinduced φ inferInstance := by
letI : TopologicalSpace B := moduleTopology R B
haveI : IsModuleTopology R B := ⟨rfl⟩
exact (isQuotientMap_of_surjective hφ).eq_coinduced | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | _root_.ModuleTopology.eq_coinduced_of_surjective | null |
instQuot (S : Submodule R A) : IsModuleTopology R (A ⧸ S) := by
constructor
have := toContinuousAdd R A
have quot := (Submodule.isOpenQuotientMap_mkQ S).isQuotientMap.eq_coinduced
have module := ModuleTopology.eq_coinduced_of_surjective <| Submodule.mkQ_surjective S
rw [quot, module] | instance | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | instQuot | null |
instProd : IsModuleTopology R (M × N) := by
constructor
have : ContinuousAdd M := toContinuousAdd R M
have : ContinuousAdd N := toContinuousAdd R N
refine le_antisymm ?_ <| sInf_le ⟨Prod.continuousSMul, Prod.continuousAdd⟩
let P := M × N
let τP : TopologicalSpace P := moduleTopology R P
have : IsModuleTop... | instance | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | instProd | The product of the module topologies for two modules over a topological ring
is the module topology. |
instPi : IsModuleTopology R (∀ i, A i) := by
induction ι using Finite.induction_empty_option
· -- invariance under equivalence of the finite type we're taking the product over
case of_equiv X Y e _ _ _ _ _ =>
exact iso (ContinuousLinearEquiv.piCongrLeft R A e)
· -- empty case
infer_instance
· -- "in... | instance | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | instPi | The product of the module topologies for a finite family of modules over a topological ring
is the module topology. |
continuous_bilinear_of_pi_fintype (ι : Type*) [Finite ι]
(bil : (ι → R) →ₗ[R] B →ₗ[R] C) : Continuous (fun ab ↦ bil ab.1 ab.2 : ((ι → R) × B → C)) := by
classical
cases nonempty_fintype ι
have h : (fun fb ↦ bil fb.1 fb.2 : ((ι → R) × B → C)) =
(fun fb ↦ ∑ i, ((fb.1 i) • (bil (Finsupp.single i 1) fb.2) :... | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | continuous_bilinear_of_pi_fintype | If `n` is finite and `B`,`C` are `R`-modules with the module topology,
then any bilinear map `Rⁿ × B → C` is automatically continuous.
Note that whilst this result works for semirings, for rings this result is superseded
by `IsModuleTopology.continuous_bilinear_of_finite_left`. |
@[continuity, fun_prop]
continuous_bilinear_of_finite_left [Module.Finite R A]
(bil : A →ₗ[R] B →ₗ[R] C) : Continuous (fun ab ↦ bil ab.1 ab.2 : (A × B → C)) := by
obtain ⟨m, f, hf⟩ := Module.Finite.exists_fin' R A
let bil' : (Fin m → R) →ₗ[R] B →ₗ[R] C := bil.comp f
let φ := f.prodMap (LinearMap.id : B →ₗ[R] ... | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | continuous_bilinear_of_finite_left | If `A`, `B` and `C` have the module topology, and if furthermore `A` is a finite `R`-module,
then any bilinear map `A × B → C` is automatically continuous |
@[continuity, fun_prop]
continuous_bilinear_of_finite_right [Module.Finite R B]
(bil : A →ₗ[R] B →ₗ[R] C) : Continuous (fun ab ↦ bil ab.1 ab.2 : (A × B → C)) := by
rw [show (fun ab ↦ bil ab.1 ab.2 : (A × B → C)) =
((fun ba ↦ bil.flip ba.1 ba.2 : (B × A → C)) ∘ (Prod.swap : A × B → B × A)) by ext; simp]
fun_... | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | continuous_bilinear_of_finite_right | If `A`, `B` and `C` have the module topology, and if furthermore `B` is a finite `R`-module,
then any bilinear map `A × B → C` is automatically continuous |
@[continuity, fun_prop]
continuous_mul_of_finite : Continuous (fun ab ↦ ab.1 * ab.2 : D × D → D) :=
continuous_bilinear_of_finite_left (LinearMap.mul R D)
include R in | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | continuous_mul_of_finite | If `D` is an `R`-algebra, finite as an `R`-module, and if `D` has the module topology,
then multiplication on `D` is automatically continuous. |
isTopologicalRing : IsTopologicalRing D where
continuous_add := (toContinuousAdd R D).1
continuous_mul := continuous_mul_of_finite R D
continuous_neg := continuous_neg R D | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.RingTheory.Finiteness.Cardinality",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Module/ModuleTopology.lean | isTopologicalRing | If `R` is a topological ring and `D` is an `R`-algebra, finite as an `R`-module,
and if `D` is given the module topology, then `D` is a topological ring. |
@[ext]
IsContPerfPair (p : M →ₗ[R] N →ₗ[R] R) where
continuous_uncurry (p) : Continuous fun (x, y) ↦ p x y
bijective_left (p) :
Bijective fun x ↦ ContinuousLinearMap.mk (p x) <| continuous_uncurry.comp <| .prodMk_right x
bijective_right (p) :
Bijective fun y ↦ ContinuousLinearMap.mk (p.flip y) <| continuo... | class | Topology | [
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/PerfectPairing.lean | IsContPerfPair | For a topological ring `R` and two topological modules `M` and `N`, a continuous perfect pairing
is a continuous bilinear map `M × N → R` that is bijective in both arguments.
We require continuity in the forward direction only so that we can put several different topologies
on the continuous dual: strong, weak, weak-*... |
flip.instIsContPerfPair : p.flip.IsContPerfPair where
continuous_uncurry := p.continuous_uncurry_of_isContPerfPair.comp continuous_swap
bijective_left := IsContPerfPair.bijective_right p
bijective_right := IsContPerfPair.bijective_left p | instance | Topology | [
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/PerfectPairing.lean | flip.instIsContPerfPair | Given a perfect pairing between `M`and `N`, we may interchange the roles of `M` and `N`. |
continuous_of_isContPerfPair : Continuous (p x) :=
p.continuous_uncurry_of_isContPerfPair.comp <| .prodMk_right x
variable [IsTopologicalRing R] | lemma | Topology | [
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/PerfectPairing.lean | continuous_of_isContPerfPair | null |
noncomputable toContPerfPair : M ≃ₗ[R] StrongDual R N :=
.ofBijective { toFun := _, map_add' x y := by ext; simp, map_smul' r x := by ext; simp } <|
IsContPerfPair.bijective_left p
@[simp] lemma toLinearMap_toContPerfPair (x : M) : p.toContPerfPair x = p x := rfl
@[simp] lemma toContPerfPair_apply (x : M) (y : N)... | def | Topology | [
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Topology.Algebra.Module.LinearMap"
] | Mathlib/Topology/Algebra/Module/PerfectPairing.lean | toContPerfPair | Turn a continuous perfect pairing between `M` and `N` into a map from `M` to continuous linear
maps `N → R`. |
perfectSpace_of_module : PerfectSpace E := by
refine ⟨fun x hx ↦ ?_⟩
let ⟨r, hr₀, hr⟩ := NormedField.exists_norm_lt_one 𝕜
obtain ⟨c, hc⟩ : ∃ (c : E), c ≠ 0 := exists_ne 0
have A : Tendsto (fun (n : ℕ) ↦ x + r ^ n • c) atTop (𝓝 (x + (0 : 𝕜) • c)) := by
apply Tendsto.add tendsto_const_nhds
exact (tends... | lemma | Topology | [
"Mathlib.Analysis.SpecificLimits.Normed",
"Mathlib.Topology.Perfect"
] | Mathlib/Topology/Algebra/Module/PerfectSpace.lean | perfectSpace_of_module | null |
LinearMap.isClosed_or_dense_ker (l : M →ₗ[R] N) :
IsClosed (LinearMap.ker l : Set M) ∨ Dense (LinearMap.ker l : Set M) := by
rcases l.surjective_or_eq_zero with (hl | rfl)
· exact (LinearMap.ker l).isClosed_or_dense_of_isCoatom (LinearMap.isCoatom_ker_of_surjective hl)
· rw [LinearMap.ker_zero]
left
e... | theorem | Topology | [
"Mathlib.RingTheory.SimpleModule.Basic",
"Mathlib.Topology.Algebra.Module.Basic"
] | Mathlib/Topology/Algebra/Module/Simple.lean | LinearMap.isClosed_or_dense_ker | The kernel of a linear map taking values in a simple module over the base ring is closed or
dense. Applies, e.g., to the case when `R = N` is a division ring. |
@[simps!]
starL (R : Type*) {A : Type*} [CommSemiring R] [StarRing R] [AddCommMonoid A]
[StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
A ≃L⋆[R] A where
toLinearEquiv := starLinearEquiv R
continuous_toFun := continuous_star
continuous_invFun := continuous_star | def | Topology | [
"Mathlib.Algebra.Star.Module",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/Module/Star.lean | starL | If `A` is a topological module over a commutative `R` with compatible actions,
then `star` is a continuous semilinear equivalence. |
@[simps!]
starL' (R : Type*) {A : Type*} [CommSemiring R] [StarRing R] [TrivialStar R] [AddCommMonoid A]
[StarAddMonoid A] [Module R A] [StarModule R A] [TopologicalSpace A] [ContinuousStar A] :
A ≃L[R] A :=
(starL R : A ≃L⋆[R] A).trans
({ AddEquiv.refl A with
map_smul' := fun r a => by simp
... | def | Topology | [
"Mathlib.Algebra.Star.Module",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/Module/Star.lean | starL' | If `A` is a topological module over a commutative `R` with trivial star and compatible actions,
then `star` is a continuous linear equivalence. |
continuous_selfAdjointPart [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] :
Continuous (selfAdjointPart R (A := A)) :=
((continuous_const_smul _).comp <| continuous_id.add continuous_star).subtype_mk _ | theorem | Topology | [
"Mathlib.Algebra.Star.Module",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/Module/Star.lean | continuous_selfAdjointPart | null |
continuous_skewAdjointPart [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] :
Continuous (skewAdjointPart R (A := A)) :=
((continuous_const_smul _).comp <| continuous_id.sub continuous_star).subtype_mk _ | theorem | Topology | [
"Mathlib.Algebra.Star.Module",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/Module/Star.lean | continuous_skewAdjointPart | null |
continuous_decomposeProdAdjoint [IsTopologicalAddGroup A] [ContinuousStar A]
[ContinuousConstSMul R A] : Continuous (StarModule.decomposeProdAdjoint R A) :=
(continuous_selfAdjointPart R A).prodMk (continuous_skewAdjointPart R A) | theorem | Topology | [
"Mathlib.Algebra.Star.Module",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/Module/Star.lean | continuous_decomposeProdAdjoint | null |
continuous_decomposeProdAdjoint_symm [ContinuousAdd A] :
Continuous (StarModule.decomposeProdAdjoint R A).symm :=
(continuous_subtype_val.comp continuous_fst).add (continuous_subtype_val.comp continuous_snd) | theorem | Topology | [
"Mathlib.Algebra.Star.Module",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/Module/Star.lean | continuous_decomposeProdAdjoint_symm | null |
@[simps! -isSimp]
selfAdjointPartL [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] :
A →L[R] selfAdjoint A where
toLinearMap := selfAdjointPart R
cont := continuous_selfAdjointPart _ _ | def | Topology | [
"Mathlib.Algebra.Star.Module",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/Module/Star.lean | selfAdjointPartL | The self-adjoint part of an element of a star module, as a continuous linear map. |
@[simps!]
skewAdjointPartL [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] :
A →L[R] skewAdjoint A where
toLinearMap := skewAdjointPart R
cont := continuous_skewAdjointPart _ _ | def | Topology | [
"Mathlib.Algebra.Star.Module",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/Module/Star.lean | skewAdjointPartL | The skew-adjoint part of an element of a star module, as a continuous linear map. |
@[simps!]
StarModule.decomposeProdAdjointL [IsTopologicalAddGroup A] [ContinuousStar A]
[ContinuousConstSMul R A] : A ≃L[R] selfAdjoint A × skewAdjoint A where
toLinearEquiv := StarModule.decomposeProdAdjoint R A
continuous_toFun := continuous_decomposeProdAdjoint _ _
continuous_invFun := continuous_decompose... | def | Topology | [
"Mathlib.Algebra.Star.Module",
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Star"
] | Mathlib/Topology/Algebra/Module/Star.lean | StarModule.decomposeProdAdjointL | The decomposition of elements of a star module into their self- and skew-adjoint parts,
as a continuous linear equivalence. |
@[nolint unusedArguments]
UniformConvergenceCLM [TopologicalSpace F] (_ : Set (Set E)) := E →SL[σ] F | def | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] | Mathlib/Topology/Algebra/Module/StrongTopology.lean | UniformConvergenceCLM | Given `E` and `F` two topological vector spaces and `𝔖 : Set (Set E)`, then
`UniformConvergenceCLM σ F 𝔖` is a type synonym of `E →SL[σ] F` equipped with the "topology of
uniform convergence on the elements of `𝔖`".
If the continuous linear image of any element of `𝔖` is bounded, this makes `E →SL[σ] F` a
topologi... |
instFunLike [TopologicalSpace F] (𝔖 : Set (Set E)) :
FunLike (UniformConvergenceCLM σ F 𝔖) E F :=
ContinuousLinearMap.funLike | instance | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] | Mathlib/Topology/Algebra/Module/StrongTopology.lean | instFunLike | null |
instContinuousSemilinearMapClass [TopologicalSpace F] (𝔖 : Set (Set E)) :
ContinuousSemilinearMapClass (UniformConvergenceCLM σ F 𝔖) σ E F :=
ContinuousLinearMap.continuousSemilinearMapClass | instance | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] | Mathlib/Topology/Algebra/Module/StrongTopology.lean | instContinuousSemilinearMapClass | null |
instTopologicalSpace [TopologicalSpace F] [IsTopologicalAddGroup F] (𝔖 : Set (Set E)) :
TopologicalSpace (UniformConvergenceCLM σ F 𝔖) :=
(@UniformOnFun.topologicalSpace E F (IsTopologicalAddGroup.toUniformSpace F) 𝔖).induced
(DFunLike.coe : (UniformConvergenceCLM σ F 𝔖) → (E →ᵤ[𝔖] F)) | instance | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.Topology.Algebra.Module.UniformConvergence",
"Mathlib.Topology.Algebra.SeparationQuotient.Section",
"Mathlib.Topology.Hom.ContinuousEvalConst"
] | Mathlib/Topology/Algebra/Module/StrongTopology.lean | instTopologicalSpace | null |
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