statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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locally_convex_space_iff_exists_convex_subset_zero :
locally_convex_space 𝕜 E ↔
∀ U ∈ (𝓝 0 : filter E), ∃ S ∈ (𝓝 0 : filter E), convex 𝕜 S ∧ S ⊆ U | (locally_convex_space_iff_zero 𝕜 E).trans has_basis_self | lemma | locally_convex_space_iff_exists_convex_subset_zero | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"convex",
"filter",
"locally_convex_space",
"locally_convex_space_iff_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_convex_space.to_locally_connected_space [module ℝ E]
[has_continuous_smul ℝ E] [locally_convex_space ℝ E] :
locally_connected_space E | locally_connected_space_of_connected_bases _ _
(λ x, @locally_convex_space.convex_basis ℝ _ _ _ _ _ _ x)
(λ x s hs, hs.2.is_preconnected) | instance | locally_convex_space.to_locally_connected_space | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"has_continuous_smul",
"locally_connected_space",
"locally_connected_space_of_connected_bases",
"locally_convex_space",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_convex_space.convex_open_basis_zero [locally_convex_space 𝕜 E] :
(𝓝 0 : filter E).has_basis (λ s, (0 : E) ∈ s ∧ is_open s ∧ convex 𝕜 s) id | (locally_convex_space.convex_basis_zero 𝕜 E).to_has_basis
(λ s hs, ⟨interior s, ⟨mem_interior_iff_mem_nhds.mpr hs.1, is_open_interior,
hs.2.interior⟩, interior_subset⟩)
(λ s hs, ⟨s, ⟨hs.2.1.mem_nhds hs.1, hs.2.2⟩, subset_rfl⟩) | lemma | locally_convex_space.convex_open_basis_zero | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"convex",
"filter",
"is_open",
"is_open_interior",
"locally_convex_space",
"locally_convex_space.convex_basis_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.exists_open_convexes [locally_convex_space 𝕜 E] {s t : set E} (disj : disjoint s t)
(hs₁ : convex 𝕜 s) (hs₂ : is_compact s) (ht₁ : convex 𝕜 t) (ht₂ : is_closed t) :
∃ u v, is_open u ∧ is_open v ∧ convex 𝕜 u ∧ convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ disjoint u v | begin
letI : uniform_space E := topological_add_group.to_uniform_space E,
haveI : uniform_add_group E := topological_add_comm_group_is_uniform,
have := (locally_convex_space.convex_open_basis_zero 𝕜 E).comap (λ x : E × E, x.2 - x.1),
rw ← uniformity_eq_comap_nhds_zero at this,
rcases disj.exists_uniform_thic... | lemma | disjoint.exists_open_convexes | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"convex",
"disjoint",
"is_closed",
"is_compact",
"is_open",
"locally_convex_space",
"locally_convex_space.convex_open_basis_zero",
"uniform_add_group",
"uniform_space",
"uniform_space.ball"
] | In a locally convex space, if `s`, `t` are disjoint convex sets, `s` is compact and `t` is
closed, then we can find open disjoint convex sets containing them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_convex_space_Inf {ts : set (topological_space E)}
(h : ∀ t ∈ ts, @locally_convex_space 𝕜 E _ _ _ t) :
@locally_convex_space 𝕜 E _ _ _ (Inf ts) | begin
letI : topological_space E := Inf ts,
refine locally_convex_space.of_bases 𝕜 E
(λ x, λ If : set ts × (ts → set E), ⋂ i ∈ If.1, If.2 i)
(λ x, λ If : set ts × (ts → set E), If.1.finite ∧ ∀ i ∈ If.1,
((If.2 i) ∈ @nhds _ ↑i x ∧ convex 𝕜 (If.2 i)))
(λ x, _) (λ x If hif, convex_Inter $ λ i, conv... | lemma | locally_convex_space_Inf | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"convex",
"convex_Inter",
"infi_subtype''",
"locally_convex_space",
"locally_convex_space.of_bases",
"locally_convex_space_iff",
"nhds",
"nhds_Inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_convex_space_infi {ts' : ι → topological_space E}
(h' : ∀ i, @locally_convex_space 𝕜 E _ _ _ (ts' i)) :
@locally_convex_space 𝕜 E _ _ _ (⨅ i, ts' i) | begin
refine locally_convex_space_Inf _,
rwa forall_range_iff
end | lemma | locally_convex_space_infi | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"locally_convex_space",
"locally_convex_space_Inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_convex_space_inf {t₁ t₂ : topological_space E}
(h₁ : @locally_convex_space 𝕜 E _ _ _ t₁) (h₂ : @locally_convex_space 𝕜 E _ _ _ t₂) :
@locally_convex_space 𝕜 E _ _ _ (t₁ ⊓ t₂) | by {rw inf_eq_infi, refine locally_convex_space_infi (λ b, _), cases b; assumption} | lemma | locally_convex_space_inf | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"inf_eq_infi",
"locally_convex_space",
"locally_convex_space_infi",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_convex_space_induced {t : topological_space F} [locally_convex_space 𝕜 F]
(f : E →ₗ[𝕜] F) :
@locally_convex_space 𝕜 E _ _ _ (t.induced f) | begin
letI : topological_space E := t.induced f,
refine locally_convex_space.of_bases 𝕜 E (λ x, preimage f)
(λ x, λ (s : set F), s ∈ 𝓝 (f x) ∧ convex 𝕜 s) (λ x, _)
(λ x s ⟨_, hs⟩, hs.linear_preimage f),
rw nhds_induced,
exact (locally_convex_space.convex_basis $ f x).comap f
end | lemma | locally_convex_space_induced | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"convex",
"locally_convex_space",
"locally_convex_space.of_bases",
"nhds_induced",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map (R : Type u) {ι : Type v} (M₁ : ι → Type w₁) (M₂ : Type w₂)
[semiring R] [∀i, add_comm_monoid (M₁ i)] [add_comm_monoid M₂]
[∀i, module R (M₁ i)] [module R M₂] [∀i, topological_space (M₁ i)] [topological_space M₂]
extends multilinear_map R M₁ M₂ | (cont : continuous to_fun) | structure | continuous_multilinear_map | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"add_comm_monoid",
"cont",
"continuous",
"module",
"multilinear_map",
"semiring",
"topological_space"
] | Continuous multilinear maps over the ring `R`, from `Πi, M₁ i` to `M₂` where `M₁ i` and `M₂`
are modules over `R` with a topological structure. In applications, there will be compatibility
conditions between the algebraic and the topological structures, but this is not needed for the
definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_multilinear_map_injective :
function.injective (continuous_multilinear_map.to_multilinear_map :
continuous_multilinear_map R M₁ M₂ → multilinear_map R M₁ M₂) | | ⟨f, hf⟩ ⟨g, hg⟩ rfl := rfl | theorem | continuous_multilinear_map.to_multilinear_map_injective | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map",
"multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_map_class :
continuous_map_class (continuous_multilinear_map R M₁ M₂) (Π i, M₁ i) M₂ | { coe := λ f, f.to_fun,
coe_injective' := λ f g h, to_multilinear_map_injective $ multilinear_map.coe_injective h,
map_continuous := continuous_multilinear_map.cont } | instance | continuous_multilinear_map.continuous_map_class | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_map_class",
"continuous_multilinear_map",
"multilinear_map.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.apply (L₁ : continuous_multilinear_map R M₁ M₂) (v : Π i, M₁ i) : M₂ | L₁ v
initialize_simps_projections continuous_multilinear_map
(-to_multilinear_map, to_multilinear_map_to_fun → apply) | def | continuous_multilinear_map.simps.apply | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_continuous : continuous (f : (Π i, M₁ i) → M₂) | f.cont | lemma | continuous_multilinear_map.coe_continuous | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe : (f.to_multilinear_map : (Π i, M₁ i) → M₂) = f | rfl | lemma | continuous_multilinear_map.coe_coe | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"coe_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f f' : continuous_multilinear_map R M₁ M₂} (H : ∀ x, f x = f' x) : f = f' | fun_like.ext _ _ H | theorem | continuous_multilinear_map.ext | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f f' : continuous_multilinear_map R M₁ M₂} : f = f' ↔ ∀ x, f x = f' x | by rw [← to_multilinear_map_injective.eq_iff, multilinear_map.ext_iff]; refl | theorem | continuous_multilinear_map.ext_iff | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map",
"multilinear_map.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coord_zero {m : Πi, M₁ i} (i : ι) (h : m i = 0) : f m = 0 | f.to_multilinear_map.map_coord_zero i h | lemma | continuous_multilinear_map.map_coord_zero | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (m : Πi, M₁ i) : (0 : continuous_multilinear_map R M₁ M₂) m = 0 | rfl | lemma | continuous_multilinear_map.zero_apply | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_multilinear_map_zero :
(0 : continuous_multilinear_map R M₁ M₂).to_multilinear_map = 0 | rfl | lemma | continuous_multilinear_map.to_multilinear_map_zero | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply (f : continuous_multilinear_map A M₁ M₂) (c : R') (m : Πi, M₁ i) :
(c • f) m = c • f m | rfl | lemma | continuous_multilinear_map.smul_apply | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_multilinear_map_smul (c : R') (f : continuous_multilinear_map A M₁ M₂) :
(c • f).to_multilinear_map = c • f.to_multilinear_map | rfl | lemma | continuous_multilinear_map.to_multilinear_map_smul | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_multilinear_map_add (f g : continuous_multilinear_map R M₁ M₂) :
(f + g).to_multilinear_map = f.to_multilinear_map + g.to_multilinear_map | rfl | lemma | continuous_multilinear_map.to_multilinear_map_add | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_monoid : add_comm_monoid (continuous_multilinear_map R M₁ M₂) | to_multilinear_map_injective.add_comm_monoid _ rfl (λ _ _, rfl) (λ _ _, rfl) | instance | continuous_multilinear_map.add_comm_monoid | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"add_comm_monoid",
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_add_hom (m : Π i, M₁ i) : continuous_multilinear_map R M₁ M₂ →+ M₂ | ⟨λ f, f m, rfl, λ _ _, rfl⟩ | def | continuous_multilinear_map.apply_add_hom | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | Evaluation of a `continuous_multilinear_map` at a vector as an `add_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_apply {α : Type*} (f : α → continuous_multilinear_map R M₁ M₂)
(m : Πi, M₁ i) {s : finset α} : (∑ a in s, f a) m = ∑ a in s, f a m | (apply_add_hom m).map_sum f s | lemma | continuous_multilinear_map.sum_apply | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_linear_map [decidable_eq ι] (m : Πi, M₁ i) (i : ι) : M₁ i →L[R] M₂ | { cont := f.cont.comp (continuous_const.update i continuous_id),
.. f.to_multilinear_map.to_linear_map m i } | def | continuous_multilinear_map.to_continuous_linear_map | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"cont",
"continuous_id"
] | If `f` is a continuous multilinear map, then `f.to_continuous_linear_map m i` is the continuous
linear map obtained by fixing all coordinates but `i` equal to those of `m`, and varying the
`i`-th coordinate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod (f : continuous_multilinear_map R M₁ M₂) (g : continuous_multilinear_map R M₁ M₃) :
continuous_multilinear_map R M₁ (M₂ × M₃) | { cont := f.cont.prod_mk g.cont,
.. f.to_multilinear_map.prod g.to_multilinear_map } | def | continuous_multilinear_map.prod | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"cont",
"continuous_multilinear_map"
] | The cartesian product of two continuous multilinear maps, as a continuous multilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_apply
(f : continuous_multilinear_map R M₁ M₂) (g : continuous_multilinear_map R M₁ M₃) (m : Πi, M₁ i) :
(f.prod g) m = (f m, g m) | rfl | lemma | continuous_multilinear_map.prod_apply | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi {ι' : Type*} {M' : ι' → Type*} [Π i, add_comm_monoid (M' i)] [Π i, topological_space (M' i)]
[Π i, module R (M' i)] (f : Π i, continuous_multilinear_map R M₁ (M' i)) :
continuous_multilinear_map R M₁ (Π i, M' i) | { cont := continuous_pi $ λ i, (f i).coe_continuous,
to_multilinear_map := multilinear_map.pi (λ i, (f i).to_multilinear_map) } | def | continuous_multilinear_map.pi | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"add_comm_monoid",
"cont",
"continuous_multilinear_map",
"continuous_pi",
"module",
"multilinear_map.pi",
"topological_space"
] | Combine a family of continuous multilinear maps with the same domain and codomains `M' i` into a
continuous multilinear map taking values in the space of functions `Π i, M' i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_pi {ι' : Type*} {M' : ι' → Type*} [Π i, add_comm_monoid (M' i)]
[Π i, topological_space (M' i)] [Π i, module R (M' i)]
(f : Π i, continuous_multilinear_map R M₁ (M' i)) :
⇑(pi f) = λ m j, f j m | rfl | lemma | continuous_multilinear_map.coe_pi | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"add_comm_monoid",
"continuous_multilinear_map",
"module",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_apply {ι' : Type*} {M' : ι' → Type*} [Π i, add_comm_monoid (M' i)]
[Π i, topological_space (M' i)] [Π i, module R (M' i)]
(f : Π i, continuous_multilinear_map R M₁ (M' i)) (m : Π i, M₁ i) (j : ι') :
pi f m j = f j m | rfl | lemma | continuous_multilinear_map.pi_apply | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"add_comm_monoid",
"continuous_multilinear_map",
"module",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cod_restrict (f : continuous_multilinear_map R M₁ M₂) (p : submodule R M₂) (h : ∀ v, f v ∈ p) :
continuous_multilinear_map R M₁ p | ⟨f.1.cod_restrict p h, f.cont.subtype_mk _⟩ | def | continuous_multilinear_map.cod_restrict | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map",
"submodule"
] | Restrict the codomain of a continuous multilinear map to a submodule. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_subsingleton [subsingleton ι] (i' : ι) : continuous_multilinear_map R (λ _ : ι, M₂) M₂ | { to_multilinear_map := multilinear_map.of_subsingleton R _ i',
cont := continuous_apply _ } | def | continuous_multilinear_map.of_subsingleton | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"cont",
"continuous_apply",
"continuous_multilinear_map",
"multilinear_map.of_subsingleton"
] | The evaluation map from `ι → M₂` to `M₂` is multilinear at a given `i` when `ι` is subsingleton. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_of_is_empty [is_empty ι] (m : M₂) : continuous_multilinear_map R M₁ M₂ | { to_multilinear_map := multilinear_map.const_of_is_empty R _ m,
cont := continuous_const } | def | continuous_multilinear_map.const_of_is_empty | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"cont",
"continuous_const",
"continuous_multilinear_map",
"is_empty",
"multilinear_map.const_of_is_empty"
] | The constant map is multilinear when `ι` is empty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_linear_map
(g : continuous_multilinear_map R M₁' M₄) (f : Π i : ι, M₁ i →L[R] M₁' i) :
continuous_multilinear_map R M₁ M₄ | { cont := g.cont.comp $ continuous_pi $ λj, (f j).cont.comp $ continuous_apply _,
.. g.to_multilinear_map.comp_linear_map (λ i, (f i).to_linear_map) } | def | continuous_multilinear_map.comp_continuous_linear_map | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"cont",
"continuous_apply",
"continuous_multilinear_map",
"continuous_pi"
] | If `g` is continuous multilinear and `f` is a collection of continuous linear maps,
then `g (f₁ m₁, ..., fₙ mₙ)` is again a continuous multilinear map, that we call
`g.comp_continuous_linear_map f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_linear_map_apply (g : continuous_multilinear_map R M₁' M₄)
(f : Π i : ι, M₁ i →L[R] M₁' i) (m : Π i, M₁ i) :
g.comp_continuous_linear_map f m = g (λ i, f i $ m i) | rfl | lemma | continuous_multilinear_map.comp_continuous_linear_map_apply | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_linear_map.comp_continuous_multilinear_map
(g : M₂ →L[R] M₃) (f : continuous_multilinear_map R M₁ M₂) :
continuous_multilinear_map R M₁ M₃ | { cont := g.cont.comp f.cont,
.. g.to_linear_map.comp_multilinear_map f.to_multilinear_map } | def | continuous_linear_map.comp_continuous_multilinear_map | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"cont",
"continuous_multilinear_map"
] | Composing a continuous multilinear map with a continuous linear map gives again a
continuous multilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.continuous_linear_map.comp_continuous_multilinear_map_coe (g : M₂ →L[R] M₃)
(f : continuous_multilinear_map R M₁ M₂) :
((g.comp_continuous_multilinear_map f) : (Πi, M₁ i) → M₃) =
(g : M₂ → M₃) ∘ (f : (Πi, M₁ i) → M₂) | by { ext m, refl } | lemma | continuous_linear_map.comp_continuous_multilinear_map_coe | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_equiv {ι' : Type*} {M' : ι' → Type*} [Π i, add_comm_monoid (M' i)]
[Π i, topological_space (M' i)] [Π i, module R (M' i)] :
(Π i, continuous_multilinear_map R M₁ (M' i)) ≃
continuous_multilinear_map R M₁ (Π i, M' i) | { to_fun := continuous_multilinear_map.pi,
inv_fun := λ f i, (continuous_linear_map.proj i : _ →L[R] M' i).comp_continuous_multilinear_map f,
left_inv := λ f, by { ext, refl },
right_inv := λ f, by { ext, refl } } | def | continuous_multilinear_map.pi_equiv | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"add_comm_monoid",
"continuous_linear_map.proj",
"continuous_multilinear_map",
"continuous_multilinear_map.pi",
"inv_fun",
"module",
"topological_space"
] | `continuous_multilinear_map.pi` as an `equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dom_dom_congr {ι' : Type*} (e : ι ≃ ι') (f : continuous_multilinear_map R (λ _ : ι, M₂) M₃) :
continuous_multilinear_map R (λ _ : ι', M₂) M₃ | { to_multilinear_map := f.dom_dom_congr e,
cont := f.cont.comp $ continuous_pi $ λ _, continuous_apply _ } | def | continuous_multilinear_map.dom_dom_congr | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"cont",
"continuous_apply",
"continuous_multilinear_map",
"continuous_pi"
] | An equivalence of the index set defines an equivalence between the spaces of continuous
multilinear maps. This is the forward map of this equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dom_dom_congr_equiv {ι' : Type*} (e : ι ≃ ι') :
continuous_multilinear_map R (λ _ : ι, M₂) M₃ ≃ continuous_multilinear_map R (λ _ : ι', M₂) M₃ | { to_fun := dom_dom_congr e,
inv_fun := dom_dom_congr e.symm,
left_inv := λ _, ext $ λ _, by simp,
right_inv := λ _, ext $ λ _, by simp } | def | continuous_multilinear_map.dom_dom_congr_equiv | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map",
"inv_fun"
] | An equivalence of the index set defines an equivalence between the spaces of continuous
multilinear maps. In case of normed spaces, this is a linear isometric equivalence, see
`continuous.multilinear_map.dom_dom_congrₗᵢ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cons_add (f : continuous_multilinear_map R M M₂) (m : Π(i : fin n), M i.succ) (x y : M 0) :
f (cons (x+y) m) = f (cons x m) + f (cons y m) | f.to_multilinear_map.cons_add m x y | lemma | continuous_multilinear_map.cons_add | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | In the specific case of continuous multilinear maps on spaces indexed by `fin (n+1)`, where one
can build an element of `Π(i : fin (n+1)), M i` using `cons`, one can express directly the
additivity of a multilinear map along the first variable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cons_smul
(f : continuous_multilinear_map R M M₂) (m : Π(i : fin n), M i.succ) (c : R) (x : M 0) :
f (cons (c • x) m) = c • f (cons x m) | f.to_multilinear_map.cons_smul m c x | lemma | continuous_multilinear_map.cons_smul | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | In the specific case of continuous multilinear maps on spaces indexed by `fin (n+1)`, where one
can build an element of `Π(i : fin (n+1)), M i` using `cons`, one can express directly the
multiplicativity of a multilinear map along the first variable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_piecewise_add [decidable_eq ι] (m m' : Πi, M₁ i) (t : finset ι) :
f (t.piecewise (m + m') m') = ∑ s in t.powerset, f (s.piecewise m m') | f.to_multilinear_map.map_piecewise_add _ _ _ | lemma | continuous_multilinear_map.map_piecewise_add | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add_univ [decidable_eq ι] [fintype ι] (m m' : Πi, M₁ i) :
f (m + m') = ∑ s : finset ι, f (s.piecewise m m') | f.to_multilinear_map.map_add_univ _ _ | lemma | continuous_multilinear_map.map_add_univ | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"finset",
"fintype"
] | Additivity of a continuous multilinear map along all coordinates at the same time,
writing `f (m + m')` as the sum of `f (s.piecewise m m')` over all sets `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_sum_finset [decidable_eq ι] :
f (λ i, ∑ j in A i, g i j) = ∑ r in pi_finset A, f (λ i, g i (r i)) | f.to_multilinear_map.map_sum_finset _ _ | lemma | continuous_multilinear_map.map_sum_finset | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [] | If `f` is continuous multilinear, then `f (Σ_{j₁ ∈ A₁} g₁ j₁, ..., Σ_{jₙ ∈ Aₙ} gₙ jₙ)` is the
sum of `f (g₁ (r 1), ..., gₙ (r n))` where `r` ranges over all functions with `r 1 ∈ A₁`, ...,
`r n ∈ Aₙ`. This follows from multilinearity by expanding successively with respect to each
coordinate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_sum [decidable_eq ι] [∀ i, fintype (α i)] :
f (λ i, ∑ j, g i j) = ∑ r : Π i, α i, f (λ i, g i (r i)) | f.to_multilinear_map.map_sum _ | lemma | continuous_multilinear_map.map_sum | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"fintype"
] | If `f` is continuous multilinear, then `f (Σ_{j₁} g₁ j₁, ..., Σ_{jₙ} gₙ jₙ)` is the sum of
`f (g₁ (r 1), ..., gₙ (r n))` where `r` ranges over all functions `r`. This follows from
multilinearity by expanding successively with respect to each coordinate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars (f : continuous_multilinear_map A M₁ M₂) :
continuous_multilinear_map R M₁ M₂ | { to_multilinear_map := f.to_multilinear_map.restrict_scalars R,
cont := f.cont } | def | continuous_multilinear_map.restrict_scalars | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"cont",
"continuous_multilinear_map",
"restrict_scalars"
] | Reinterpret an `A`-multilinear map as an `R`-multilinear map, if `A` is an algebra over `R`
and their actions on all involved modules agree with the action of `R` on `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_restrict_scalars (f : continuous_multilinear_map A M₁ M₂) :
⇑(f.restrict_scalars R) = f | rfl | lemma | continuous_multilinear_map.coe_restrict_scalars | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub [decidable_eq ι] (m : Πi, M₁ i) (i : ι) (x y : M₁ i) :
f (update m i (x - y)) = f (update m i x) - f (update m i y) | f.to_multilinear_map.map_sub _ _ _ _ | lemma | continuous_multilinear_map.map_sub | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_apply (m : Πi, M₁ i) : (f - f') m = f m - f' m | rfl | lemma | continuous_multilinear_map.sub_apply | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_piecewise_smul [decidable_eq ι] (c : ι → R) (m : Πi, M₁ i) (s : finset ι) :
f (s.piecewise (λ i, c i • m i) m) = (∏ i in s, c i) • f m | f.to_multilinear_map.map_piecewise_smul _ _ _ | lemma | continuous_multilinear_map.map_piecewise_smul | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul_univ [fintype ι] (c : ι → R) (m : Πi, M₁ i) :
f (λ i, c i • m i) = (∏ i, c i) • f m | f.to_multilinear_map.map_smul_univ _ _ | lemma | continuous_multilinear_map.map_smul_univ | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"fintype"
] | Multiplicativity of a continuous multilinear map along all coordinates at the same time,
writing `f (λ i, c i • m i)` as `(∏ i, c i) • f m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_multilinear_map_linear :
continuous_multilinear_map A M₁ M₂ →ₗ[R'] multilinear_map A M₁ M₂ | { to_fun := to_multilinear_map,
map_add' := to_multilinear_map_add,
map_smul' := to_multilinear_map_smul } | def | continuous_multilinear_map.to_multilinear_map_linear | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map",
"multilinear_map"
] | Linear map version of the map `to_multilinear_map` associating to a continuous multilinear map
the corresponding multilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_linear_equiv {ι' : Type*} {M' : ι' → Type*}
[Π i, add_comm_monoid (M' i)] [Π i, topological_space (M' i)] [∀ i, has_continuous_add (M' i)]
[Π i, module R' (M' i)] [Π i, module A (M' i)] [∀ i, smul_comm_class A R' (M' i)]
[Π i, has_continuous_const_smul R' (M' i)] :
(Π i, continuous_multilinear_map A M₁ (M' i... | { map_add' := λ x y, rfl,
map_smul' := λ c x, rfl,
.. pi_equiv } | def | continuous_multilinear_map.pi_linear_equiv | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"add_comm_monoid",
"continuous_multilinear_map",
"has_continuous_add",
"has_continuous_const_smul",
"module",
"smul_comm_class",
"topological_space"
] | `continuous_multilinear_map.pi` as a `linear_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_pi_algebra : continuous_multilinear_map R (λ i : ι, A) A | { cont := continuous_finset_prod _ $ λ i hi, continuous_apply _,
to_multilinear_map := multilinear_map.mk_pi_algebra R ι A} | def | continuous_multilinear_map.mk_pi_algebra | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"cont",
"continuous_apply",
"continuous_finset_prod",
"continuous_multilinear_map",
"multilinear_map.mk_pi_algebra"
] | The continuous multilinear map on `A^ι`, where `A` is a normed commutative algebra
over `𝕜`, associating to `m` the product of all the `m i`.
See also `continuous_multilinear_map.mk_pi_algebra_fin`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_pi_algebra_apply (m : ι → A) :
continuous_multilinear_map.mk_pi_algebra R ι A m = ∏ i, m i | rfl | lemma | continuous_multilinear_map.mk_pi_algebra_apply | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map.mk_pi_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_pi_algebra_fin : A [×n]→L[R] A | { cont := begin
change continuous (λ m, (list.of_fn m).prod),
simp_rw list.of_fn_eq_map,
exact continuous_list_prod _ (λ i hi, continuous_apply _),
end,
to_multilinear_map := multilinear_map.mk_pi_algebra_fin R n A} | def | continuous_multilinear_map.mk_pi_algebra_fin | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"cont",
"continuous",
"continuous_apply",
"continuous_list_prod",
"list.of_fn",
"list.of_fn_eq_map",
"multilinear_map.mk_pi_algebra_fin"
] | The continuous multilinear map on `A^n`, where `A` is a normed algebra over `𝕜`, associating to
`m` the product of all the `m i`.
See also: `continuous_multilinear_map.mk_pi_algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_pi_algebra_fin_apply (m : fin n → A) :
continuous_multilinear_map.mk_pi_algebra_fin R n A m = (list.of_fn m).prod | rfl | lemma | continuous_multilinear_map.mk_pi_algebra_fin_apply | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"continuous_multilinear_map.mk_pi_algebra_fin",
"list.of_fn"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_right : continuous_multilinear_map R M₁ M₂ | { to_multilinear_map := f.to_multilinear_map.smul_right z,
cont := f.cont.smul continuous_const } | def | continuous_multilinear_map.smul_right | topology.algebra.module | src/topology/algebra/module/multilinear.lean | [
"topology.algebra.module.basic",
"linear_algebra.multilinear.basic"
] | [
"cont",
"continuous_const",
"continuous_multilinear_map"
] | Given a continuous `R`-multilinear map `f` taking values in `R`, `f.smul_right z` is the
continuous multilinear map sending `m` to `f m • z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.is_closed_or_dense_ker (l : M →ₗ[R] N) :
is_closed (l.ker : set M) ∨ dense (l.ker : set M) | begin
rcases l.surjective_or_eq_zero with (hl|rfl),
{ exact l.ker.is_closed_or_dense_of_is_coatom (linear_map.is_coatom_ker_of_surjective hl) },
{ rw linear_map.ker_zero,
left,
exact is_closed_univ },
end | lemma | linear_map.is_closed_or_dense_ker | topology.algebra.module | src/topology/algebra/module/simple.lean | [
"ring_theory.simple_module",
"topology.algebra.module.basic"
] | [
"dense",
"is_closed",
"is_closed_univ",
"linear_map.is_coatom_ker_of_surjective",
"linear_map.ker_zero"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
starL (R : Type*) {A : Type*}
[comm_semiring R] [star_ring R] [add_comm_monoid A] [star_add_monoid A] [module R A]
[star_module R A] [topological_space A] [has_continuous_star A] :
A ≃L⋆[R] A | { to_linear_equiv := star_linear_equiv R,
continuous_to_fun := continuous_star,
continuous_inv_fun := continuous_star } | def | starL | topology.algebra.module | src/topology/algebra/module/star.lean | [
"algebra.star.module",
"topology.algebra.module.basic",
"topology.algebra.star"
] | [
"add_comm_monoid",
"comm_semiring",
"has_continuous_star",
"module",
"star_add_monoid",
"star_linear_equiv",
"star_module",
"star_ring",
"topological_space"
] | If `A` is a topological module over a commutative `R` with compatible actions,
then `star` is a continuous semilinear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
starL' (R : Type*) {A : Type*}
[comm_semiring R] [star_ring R] [has_trivial_star R] [add_comm_monoid A] [star_add_monoid A]
[module R A] [star_module R A] [topological_space A] [has_continuous_star A] :
A ≃L[R] A | (starL R : A ≃L⋆[R] A).trans
({ map_smul' := λ r a, by simp [star_ring_end_apply],
continuous_to_fun := continuous_id,
continuous_inv_fun := continuous_id,
..add_equiv.refl A, } : A ≃L⋆[R] A) | def | starL' | topology.algebra.module | src/topology/algebra/module/star.lean | [
"algebra.star.module",
"topology.algebra.module.basic",
"topology.algebra.star"
] | [
"add_comm_monoid",
"comm_semiring",
"continuous_id",
"has_continuous_star",
"has_trivial_star",
"module",
"starL",
"star_add_monoid",
"star_module",
"star_ring",
"star_ring_end_apply",
"topological_space"
] | If `A` is a topological module over a commutative `R` with trivial star and compatible actions,
then `star` is a continuous linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_self_adjoint_part [has_continuous_add A] [has_continuous_star A]
[has_continuous_const_smul R A] :
continuous (@self_adjoint_part R A _ _ _ _ _ _ _ _) | ((continuous_const_smul _).comp $ continuous_id.add continuous_star).subtype_mk _ | lemma | continuous_self_adjoint_part | topology.algebra.module | src/topology/algebra/module/star.lean | [
"algebra.star.module",
"topology.algebra.module.basic",
"topology.algebra.star"
] | [
"continuous",
"has_continuous_add",
"has_continuous_const_smul",
"has_continuous_star",
"self_adjoint_part"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_skew_adjoint_part [has_continuous_sub A] [has_continuous_star A]
[has_continuous_const_smul R A] :
continuous (@skew_adjoint_part R A _ _ _ _ _ _ _ _) | ((continuous_const_smul _).comp $ continuous_id.sub continuous_star).subtype_mk _ | lemma | continuous_skew_adjoint_part | topology.algebra.module | src/topology/algebra/module/star.lean | [
"algebra.star.module",
"topology.algebra.module.basic",
"topology.algebra.star"
] | [
"continuous",
"has_continuous_const_smul",
"has_continuous_star",
"has_continuous_sub",
"skew_adjoint_part"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_decompose_prod_adjoint [topological_add_group A] [has_continuous_star A]
[has_continuous_const_smul R A] :
continuous (@star_module.decompose_prod_adjoint R A _ _ _ _ _ _ _ _) | (continuous_self_adjoint_part R A).prod_mk (continuous_skew_adjoint_part R A) | lemma | continuous_decompose_prod_adjoint | topology.algebra.module | src/topology/algebra/module/star.lean | [
"algebra.star.module",
"topology.algebra.module.basic",
"topology.algebra.star"
] | [
"continuous",
"continuous_self_adjoint_part",
"continuous_skew_adjoint_part",
"has_continuous_const_smul",
"has_continuous_star",
"star_module.decompose_prod_adjoint",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_decompose_prod_adjoint_symm [topological_add_group A] :
continuous (@star_module.decompose_prod_adjoint R A _ _ _ _ _ _ _ _).symm | (continuous_subtype_coe.comp continuous_fst).add (continuous_subtype_coe.comp continuous_snd) | lemma | continuous_decompose_prod_adjoint_symm | topology.algebra.module | src/topology/algebra/module/star.lean | [
"algebra.star.module",
"topology.algebra.module.basic",
"topology.algebra.star"
] | [
"continuous",
"continuous_fst",
"continuous_snd",
"star_module.decompose_prod_adjoint",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_adjoint_partL [has_continuous_add A] [has_continuous_star A]
[has_continuous_const_smul R A] : A →L[R] self_adjoint A | { to_linear_map := self_adjoint_part R,
cont := continuous_self_adjoint_part _ _ } | def | self_adjoint_partL | topology.algebra.module | src/topology/algebra/module/star.lean | [
"algebra.star.module",
"topology.algebra.module.basic",
"topology.algebra.star"
] | [
"cont",
"continuous_self_adjoint_part",
"has_continuous_add",
"has_continuous_const_smul",
"has_continuous_star",
"self_adjoint",
"self_adjoint_part"
] | The self-adjoint part of an element of a star module, as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skew_adjoint_partL [has_continuous_sub A] [has_continuous_star A]
[has_continuous_const_smul R A] : A →L[R] skew_adjoint A | { to_linear_map := skew_adjoint_part R,
cont := continuous_skew_adjoint_part _ _ } | def | skew_adjoint_partL | topology.algebra.module | src/topology/algebra/module/star.lean | [
"algebra.star.module",
"topology.algebra.module.basic",
"topology.algebra.star"
] | [
"cont",
"continuous_skew_adjoint_part",
"has_continuous_const_smul",
"has_continuous_star",
"has_continuous_sub",
"skew_adjoint",
"skew_adjoint_part"
] | The skew-adjoint part of an element of a star module, as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_module.decompose_prod_adjointL [topological_add_group A] [has_continuous_star A]
[has_continuous_const_smul R A] :
A ≃L[R] self_adjoint A × skew_adjoint A | { to_linear_equiv := star_module.decompose_prod_adjoint R A,
continuous_to_fun := continuous_decompose_prod_adjoint _ _,
continuous_inv_fun := continuous_decompose_prod_adjoint_symm _ _ } | def | star_module.decompose_prod_adjointL | topology.algebra.module | src/topology/algebra/module/star.lean | [
"algebra.star.module",
"topology.algebra.module.basic",
"topology.algebra.star"
] | [
"continuous_decompose_prod_adjoint",
"continuous_decompose_prod_adjoint_symm",
"has_continuous_const_smul",
"has_continuous_star",
"self_adjoint",
"skew_adjoint",
"star_module.decompose_prod_adjoint",
"topological_add_group"
] | The decomposition of elements of a star module into their self- and skew-adjoint parts,
as a continuous linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_topology [topological_space F] [topological_add_group F]
(𝔖 : set (set E)) : topological_space (E →SL[σ] F) | (@uniform_on_fun.topological_space E F
(topological_add_group.to_uniform_space F) 𝔖).induced coe_fn | def | continuous_linear_map.strong_topology | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"topological_add_group",
"topological_space"
] | Given `E` and `F` two topological vector spaces and `𝔖 : set (set E)`, then
`strong_topology σ F 𝔖` is the "topology of uniform convergence on the elements of `𝔖`" on
`E →L[𝕜] F`.
If the continuous linear image of any element of `𝔖` is bounded, this makes `E →L[𝕜] F` a
topological vector space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_uniformity [uniform_space F] [uniform_add_group F]
(𝔖 : set (set E)) : uniform_space (E →SL[σ] F) | @uniform_space.replace_topology _ (strong_topology σ F 𝔖)
((uniform_on_fun.uniform_space E F 𝔖).comap coe_fn)
(by rw [strong_topology, uniform_add_group.to_uniform_space_eq]; refl) | def | continuous_linear_map.strong_uniformity | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"uniform_add_group",
"uniform_space",
"uniform_space.replace_topology"
] | The uniform structure associated with `continuous_linear_map.strong_topology`. We make sure
that this has nice definitional properties. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_uniformity_topology_eq [uniform_space F] [uniform_add_group F]
(𝔖 : set (set E)) :
(strong_uniformity σ F 𝔖).to_topological_space = strong_topology σ F 𝔖 | rfl | lemma | continuous_linear_map.strong_uniformity_topology_eq | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"uniform_add_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_uniformity.uniform_embedding_coe_fn [uniform_space F] [uniform_add_group F]
(𝔖 : set (set E)) :
@uniform_embedding (E →SL[σ] F) (E →ᵤ[𝔖] F) (strong_uniformity σ F 𝔖)
(uniform_on_fun.uniform_space E F 𝔖) coe_fn | begin
letI : uniform_space (E →SL[σ] F) := strong_uniformity σ F 𝔖,
exact ⟨⟨rfl⟩, fun_like.coe_injective⟩
end | lemma | continuous_linear_map.strong_uniformity.uniform_embedding_coe_fn | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"uniform_add_group",
"uniform_embedding",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_topology.embedding_coe_fn [uniform_space F] [uniform_add_group F]
(𝔖 : set (set E)) :
@embedding (E →SL[σ] F) (E →ᵤ[𝔖] F) (strong_topology σ F 𝔖)
(uniform_on_fun.topological_space E F 𝔖)
(uniform_on_fun.of_fun 𝔖 ∘ coe_fn) | @uniform_embedding.embedding _ _ (_root_.id _) _ _
(strong_uniformity.uniform_embedding_coe_fn _ _ _) | lemma | continuous_linear_map.strong_topology.embedding_coe_fn | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"embedding",
"uniform_add_group",
"uniform_embedding.embedding",
"uniform_on_fun.of_fun",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_uniformity.uniform_add_group [uniform_space F] [uniform_add_group F]
(𝔖 : set (set E)) : @uniform_add_group (E →SL[σ] F) (strong_uniformity σ F 𝔖) _ | begin
letI : uniform_space (E →SL[σ] F) := strong_uniformity σ F 𝔖,
rw [strong_uniformity, uniform_space.replace_topology_eq],
let φ : (E →SL[σ] F) →+ E →ᵤ[𝔖] F := ⟨(coe_fn : (E →SL[σ] F) → E →ᵤ F), rfl, λ _ _, rfl⟩,
exact uniform_add_group_comap φ
end | lemma | continuous_linear_map.strong_uniformity.uniform_add_group | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"uniform_add_group",
"uniform_space",
"uniform_space.replace_topology_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_topology.topological_add_group [topological_space F] [topological_add_group F]
(𝔖 : set (set E)) : @topological_add_group (E →SL[σ] F) (strong_topology σ F 𝔖) _ | begin
letI : uniform_space F := topological_add_group.to_uniform_space F,
haveI : uniform_add_group F := topological_add_comm_group_is_uniform,
letI : uniform_space (E →SL[σ] F) := strong_uniformity σ F 𝔖,
haveI : uniform_add_group (E →SL[σ] F) := strong_uniformity.uniform_add_group σ F 𝔖,
apply_instance
en... | lemma | continuous_linear_map.strong_topology.topological_add_group | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"topological_add_group",
"topological_space",
"uniform_add_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_topology.t2_space [topological_space F] [topological_add_group F] [t2_space F]
(𝔖 : set (set E)) (h𝔖 : ⋃₀ 𝔖 = set.univ) : @t2_space (E →SL[σ] F) (strong_topology σ F 𝔖) | begin
letI : uniform_space F := topological_add_group.to_uniform_space F,
haveI : uniform_add_group F := topological_add_comm_group_is_uniform,
letI : topological_space (E →SL[σ] F) := strong_topology σ F 𝔖,
haveI : t2_space (E →ᵤ[𝔖] F) := uniform_on_fun.t2_space_of_covering h𝔖,
exact (strong_topology.embe... | lemma | continuous_linear_map.strong_topology.t2_space | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"t2_space",
"topological_add_group",
"topological_space",
"uniform_add_group",
"uniform_on_fun.t2_space_of_covering",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_topology.has_continuous_smul [ring_hom_surjective σ] [ring_hom_isometric σ]
[topological_space F] [topological_add_group F] [has_continuous_smul 𝕜₂ F] (𝔖 : set (set E))
(h𝔖₁ : 𝔖.nonempty) (h𝔖₂ : directed_on (⊆) 𝔖) (h𝔖₃ : ∀ S ∈ 𝔖, bornology.is_vonN_bounded 𝕜₁ S) :
@has_continuous_smul 𝕜₂ (E →SL[σ]... | begin
letI : uniform_space F := topological_add_group.to_uniform_space F,
haveI : uniform_add_group F := topological_add_comm_group_is_uniform,
letI : topological_space (E →SL[σ] F) := strong_topology σ F 𝔖,
let φ : (E →SL[σ] F) →ₗ[𝕜₂] E →ᵤ[𝔖] F :=
⟨(coe_fn : (E →SL[σ] F) → E → F), λ _ _, rfl, λ _ _, rfl... | lemma | continuous_linear_map.strong_topology.has_continuous_smul | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"bornology.is_vonN_bounded",
"directed_on",
"has_continuous_smul",
"ring_hom_isometric",
"ring_hom_surjective",
"topological_add_group",
"topological_space",
"uniform_add_group",
"uniform_on_fun.has_continuous_smul_induced_of_image_bounded",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_topology.has_basis_nhds_zero_of_basis [topological_space F] [topological_add_group F]
{ι : Type*} (𝔖 : set (set E)) (h𝔖₁ : 𝔖.nonempty) (h𝔖₂ : directed_on (⊆) 𝔖) {p : ι → Prop}
{b : ι → set F} (h : (𝓝 0 : filter F).has_basis p b) :
(@nhds (E →SL[σ] F) (strong_topology σ F 𝔖) 0).has_basis
(λ Si : ... | begin
letI : uniform_space F := topological_add_group.to_uniform_space F,
haveI : uniform_add_group F := topological_add_comm_group_is_uniform,
rw nhds_induced,
exact (uniform_on_fun.has_basis_nhds_zero_of_basis 𝔖 h𝔖₁ h𝔖₂ h).comap coe_fn
end | lemma | continuous_linear_map.strong_topology.has_basis_nhds_zero_of_basis | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"directed_on",
"filter",
"nhds",
"nhds_induced",
"topological_add_group",
"topological_space",
"uniform_add_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strong_topology.has_basis_nhds_zero [topological_space F] [topological_add_group F]
(𝔖 : set (set E)) (h𝔖₁ : 𝔖.nonempty) (h𝔖₂ : directed_on (⊆) 𝔖) :
(@nhds (E →SL[σ] F) (strong_topology σ F 𝔖) 0).has_basis
(λ SV : set E × set F, SV.1 ∈ 𝔖 ∧ SV.2 ∈ (𝓝 0 : filter F))
(λ SV, {f : E →SL[σ] F | ∀ x ∈ SV.1... | strong_topology.has_basis_nhds_zero_of_basis σ F 𝔖 h𝔖₁ h𝔖₂ (𝓝 0).basis_sets | lemma | continuous_linear_map.strong_topology.has_basis_nhds_zero | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"directed_on",
"filter",
"nhds",
"topological_add_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_nhds_zero_of_basis [topological_space F]
[topological_add_group F] {ι : Type*} {p : ι → Prop} {b : ι → set F}
(h : (𝓝 0 : filter F).has_basis p b) :
(𝓝 (0 : E →SL[σ] F)).has_basis
(λ Si : set E × ι, bornology.is_vonN_bounded 𝕜₁ Si.1 ∧ p Si.2)
(λ Si, {f : E →SL[σ] F | ∀ x ∈ Si.1, f x ∈ b Si.2}... | strong_topology.has_basis_nhds_zero_of_basis σ F
{S | bornology.is_vonN_bounded 𝕜₁ S} ⟨∅, bornology.is_vonN_bounded_empty 𝕜₁ E⟩
(directed_on_of_sup_mem $ λ _ _, bornology.is_vonN_bounded.union) h | lemma | continuous_linear_map.has_basis_nhds_zero_of_basis | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"bornology.is_vonN_bounded",
"bornology.is_vonN_bounded.union",
"bornology.is_vonN_bounded_empty",
"directed_on_of_sup_mem",
"filter",
"topological_add_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_nhds_zero [topological_space F]
[topological_add_group F] :
(𝓝 (0 : E →SL[σ] F)).has_basis
(λ SV : set E × set F, bornology.is_vonN_bounded 𝕜₁ SV.1 ∧ SV.2 ∈ (𝓝 0 : filter F))
(λ SV, {f : E →SL[σ] F | ∀ x ∈ SV.1, f x ∈ SV.2}) | continuous_linear_map.has_basis_nhds_zero_of_basis (𝓝 0).basis_sets | lemma | continuous_linear_map.has_basis_nhds_zero | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"bornology.is_vonN_bounded",
"continuous_linear_map.has_basis_nhds_zero_of_basis",
"filter",
"topological_add_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
precomp [topological_add_group G] [has_continuous_const_smul 𝕜₃ G]
[ring_hom_surjective σ] [ring_hom_isometric σ] (L : E →SL[σ] F) :
(F →SL[τ] G) →L[𝕜₃] (E →SL[ρ] G) | { to_fun := λ f, f.comp L,
map_add' := λ f g, add_comp f g L,
map_smul' := λ a f, smul_comp a f L,
cont :=
begin
letI : uniform_space G := topological_add_group.to_uniform_space G,
haveI : uniform_add_group G := topological_add_comm_group_is_uniform,
rw (strong_topology.embedding_coe_fn _ _ _).conti... | def | continuous_linear_map.precomp | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"cont",
"continuous",
"continuous.comp",
"has_continuous_const_smul",
"ring_hom_isometric",
"ring_hom_surjective",
"topological_add_group",
"uniform_add_group",
"uniform_on_fun.precomp_uniform_continuous",
"uniform_space"
] | Pre-composition by a *fixed* continuous linear map as a continuous linear map.
Note that in non-normed space it is not always true that composition is continuous
in both variables, so we have to fix one of them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcomp [topological_add_group F] [topological_add_group G]
[has_continuous_const_smul 𝕜₃ G] [has_continuous_const_smul 𝕜₂ F] (L : F →SL[τ] G) :
(E →SL[σ] F) →SL[τ] (E →SL[ρ] G) | { to_fun := λ f, L.comp f,
map_add' := comp_add L,
map_smul' := comp_smulₛₗ L,
cont :=
begin
letI : uniform_space G := topological_add_group.to_uniform_space G,
haveI : uniform_add_group G := topological_add_comm_group_is_uniform,
letI : uniform_space F := topological_add_group.to_uniform_space F,
... | def | continuous_linear_map.postcomp | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"cont",
"continuous",
"continuous.comp",
"has_continuous_const_smul",
"topological_add_group",
"uniform_add_group",
"uniform_on_fun.postcomp_uniform_continuous",
"uniform_space"
] | Post-composition by a *fixed* continuous linear map as a continuous linear map.
Note that in non-normed space it is not always true that composition is continuous
in both variables, so we have to fix one of them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arrow_congrSL (e₁₂ : E ≃SL[σ₁₂] F) (e₄₃ : H ≃SL[σ₄₃] G) :
(E →SL[σ₁₄] H) ≃SL[σ₄₃] (F →SL[σ₂₃] G) | { -- given explicitly to help `simps`
to_fun := λ L, (e₄₃ : H →SL[σ₄₃] G).comp (L.comp (e₁₂.symm : F →SL[σ₂₁] E)),
-- given explicitly to help `simps`
inv_fun := λ L, (e₄₃.symm : G →SL[σ₃₄] H).comp (L.comp (e₁₂ : E →SL[σ₁₂] F)),
map_add' := λ f g, by rw [add_comp, comp_add],
map_smul' := λ t f, by rw [smul_co... | def | continuous_linear_equiv.arrow_congrSL | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [
"continuous",
"inv_fun"
] | A pair of continuous (semi)linear equivalences generates a (semi)linear equivalence between the
spaces of continuous (semi)linear maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arrow_congr (e₁ : E ≃L[𝕜] F) (e₂ : H ≃L[𝕜] G) : (E →L[𝕜] H) ≃L[𝕜] (F →L[𝕜] G) | e₁.arrow_congrSL e₂ | def | continuous_linear_equiv.arrow_congr | topology.algebra.module | src/topology/algebra/module/strong_topology.lean | [
"topology.algebra.uniform_convergence"
] | [] | A pair of continuous linear equivalences generates an continuous linear equivalence between
the spaces of continuous linear maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weak_bilin [comm_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] [add_comm_monoid F]
[module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) | E | def | weak_bilin | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"add_comm_monoid",
"comm_semiring",
"module"
] | The space `E` equipped with the weak topology induced by the bilinear form `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module' [comm_semiring 𝕜] [comm_semiring 𝕝] [add_comm_group E] [module 𝕜 E]
[add_comm_group F] [module 𝕜 F] [m : module 𝕝 E] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
module 𝕝 (weak_bilin B) | m | instance | weak_bilin.module' | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"add_comm_group",
"comm_semiring",
"module",
"weak_bilin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_continuous : continuous (λ (x : weak_bilin B) y, B x y) | continuous_induced_dom | lemma | weak_bilin.coe_fn_continuous | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"continuous",
"continuous_induced_dom",
"weak_bilin"
] | The coercion `(λ x y, B x y) : E → (F → 𝕜)` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_continuous (y : F) : continuous (λ x : weak_bilin B, B x y) | ( continuous_pi_iff.mp (coe_fn_continuous B)) y | lemma | weak_bilin.eval_continuous | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"continuous",
"weak_bilin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_continuous_eval [topological_space α] {g : α → weak_bilin B}
(h : ∀ y, continuous (λ a, B (g a) y)) : continuous g | continuous_induced_rng.2 (continuous_pi_iff.mpr h) | lemma | weak_bilin.continuous_of_continuous_eval | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"continuous",
"topological_space",
"weak_bilin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} (hB : function.injective B) :
embedding (λ (x : weak_bilin B) y, B x y) | function.injective.embedding_induced $ linear_map.coe_injective.comp hB | lemma | weak_bilin.embedding | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"embedding",
"function.injective.embedding_induced",
"weak_bilin"
] | The coercion `(λ x y, B x y) : E → (F → 𝕜)` is an embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_iff_forall_eval_tendsto {l : filter α} {f : α → (weak_bilin B)} {x : weak_bilin B}
(hB : function.injective B) : tendsto f l (𝓝 x) ↔ ∀ y, tendsto (λ i, B (f i) y) l (𝓝 (B x y)) | by rw [← tendsto_pi_nhds, embedding.tendsto_nhds_iff (embedding hB)] | theorem | weak_bilin.tendsto_iff_forall_eval_tendsto | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"embedding",
"embedding.tendsto_nhds_iff",
"filter",
"tendsto_pi_nhds",
"weak_bilin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_dual_pairing (𝕜 E) [comm_semiring 𝕜] [topological_space 𝕜] [has_continuous_add 𝕜]
[add_comm_monoid E] [module 𝕜 E] [topological_space E]
[has_continuous_const_smul 𝕜 𝕜] :
(E →L[𝕜] 𝕜) →ₗ[𝕜] E →ₗ[𝕜] 𝕜 | continuous_linear_map.coe_lm 𝕜 | def | top_dual_pairing | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"add_comm_monoid",
"comm_semiring",
"continuous_linear_map.coe_lm",
"has_continuous_add",
"has_continuous_const_smul",
"module",
"topological_space"
] | The canonical pairing of a vector space and its topological dual. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_pairing_apply (v : (E →L[𝕜] 𝕜)) (x : E) : top_dual_pairing 𝕜 E v x = v x | rfl | lemma | dual_pairing_apply | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"top_dual_pairing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weak_dual (𝕜 E) [comm_semiring 𝕜] [topological_space 𝕜] [has_continuous_add 𝕜]
[has_continuous_const_smul 𝕜 𝕜] [add_comm_monoid E] [module 𝕜 E] [topological_space E] | weak_bilin (top_dual_pairing 𝕜 E) | def | weak_dual | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"add_comm_monoid",
"comm_semiring",
"has_continuous_add",
"has_continuous_const_smul",
"module",
"top_dual_pairing",
"topological_space",
"weak_bilin"
] | The weak star topology is the topology coarsest topology on `E →L[𝕜] 𝕜` such that all
functionals `λ v, top_dual_pairing 𝕜 E v x` are continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weak_dual.continuous_linear_map_class :
continuous_linear_map_class (weak_dual 𝕜 E) 𝕜 E 𝕜 | continuous_linear_map.continuous_semilinear_map_class | instance | weak_dual.weak_dual.continuous_linear_map_class | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"continuous_linear_map_class",
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module' (R) [semiring R] [module R 𝕜] [smul_comm_class 𝕜 R 𝕜]
[has_continuous_const_smul R 𝕜] :
module R (weak_dual 𝕜 E) | continuous_linear_map.module | instance | weak_dual.module' | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"has_continuous_const_smul",
"module",
"semiring",
"smul_comm_class",
"weak_dual"
] | If `𝕜` is a topological module over a semiring `R` and scalar multiplication commutes with the
multiplication on `𝕜`, then `weak_dual 𝕜 E` is a module over `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_continuous : continuous (λ (x : weak_dual 𝕜 E) y, x y) | continuous_induced_dom | lemma | weak_dual.coe_fn_continuous | topology.algebra.module | src/topology/algebra/module/weak_dual.lean | [
"topology.algebra.module.basic",
"linear_algebra.bilinear_map"
] | [
"continuous",
"continuous_induced_dom",
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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