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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
fun_unique : (ι → M) ≃L[R] M | { to_linear_equiv := linear_equiv.fun_unique ι R M,
.. homeomorph.fun_unique ι M } | def | continuous_linear_equiv.fun_unique | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"homeomorph.fun_unique",
"linear_equiv.fun_unique"
] | If `ι` has a unique element, then `ι → M` is continuously linear equivalent to `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fun_unique : ⇑(fun_unique ι R M) = function.eval default | rfl | lemma | continuous_linear_equiv.coe_fun_unique | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"function.eval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fun_unique_symm : ⇑(fun_unique ι R M).symm = function.const ι | rfl | lemma | continuous_linear_equiv.coe_fun_unique_symm | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_fin_two (M : fin 2 → Type*) [Π i, add_comm_monoid (M i)] [Π i, module R (M i)]
[Π i, topological_space (M i)] :
(Π i, M i) ≃L[R] M 0 × M 1 | { to_linear_equiv := linear_equiv.pi_fin_two R M, .. homeomorph.pi_fin_two M } | def | continuous_linear_equiv.pi_fin_two | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_monoid",
"homeomorph.pi_fin_two",
"linear_equiv.pi_fin_two",
"module",
"topological_space"
] | Continuous linear equivalence between dependent functions `Π i : fin 2, M i` and `M 0 × M 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fin_two_arrow : (fin 2 → M) ≃L[R] M × M | { to_linear_equiv := linear_equiv.fin_two_arrow R M, .. pi_fin_two R (λ _, M) } | def | continuous_linear_equiv.fin_two_arrow | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"linear_equiv.fin_two_arrow"
] | Continuous linear equivalence between vectors in `M² = fin 2 → M` and `M × M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse : (M →L[R] M₂) → (M₂ →L[R] M) | λ f, if h : ∃ (e : M ≃L[R] M₂), (e : M →L[R] M₂) = f then ((classical.some h).symm : M₂ →L[R] M)
else 0 | def | continuous_linear_map.inverse | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | Introduce a function `inverse` from `M →L[R] M₂` to `M₂ →L[R] M`, which sends `f` to `f.symm` if
`f` is a continuous linear equivalence and to `0` otherwise. This definition is somewhat ad hoc,
but one needs a fully (rather than partially) defined inverse function for some purposes, including
for calculus. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse_equiv (e : M ≃L[R] M₂) : inverse (e : M →L[R] M₂) = e.symm | begin
have h : ∃ (e' : M ≃L[R] M₂), (e' : M →L[R] M₂) = ↑e := ⟨e, rfl⟩,
simp only [inverse, dif_pos h],
congr,
exact_mod_cast (classical.some_spec h)
end | lemma | continuous_linear_map.inverse_equiv | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | By definition, if `f` is invertible then `inverse f = f.symm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse_non_equiv (f : M →L[R] M₂) (h : ¬∃ (e' : M ≃L[R] M₂), ↑e' = f) :
inverse f = 0 | dif_neg h | lemma | continuous_linear_map.inverse_non_equiv | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [] | By definition, if `f` is not invertible then `inverse f = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_inverse_equiv (e : M ≃L[R] M) :
ring.inverse ↑e = inverse (e : M →L[R] M) | begin
suffices :
ring.inverse ((((continuous_linear_equiv.units_equiv _ _).symm e) : M →L[R] M)) = inverse ↑e,
{ convert this },
simp,
refl,
end | lemma | continuous_linear_map.ring_inverse_equiv | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_linear_equiv.units_equiv",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_inverse (e : M ≃L[R] M₂) (f : M →L[R] M₂) :
inverse f = (ring.inverse ((e.symm : (M₂ →L[R] M)).comp f)) ∘L ↑e.symm | begin
by_cases h₁ : ∃ (e' : M ≃L[R] M₂), ↑e' = f,
{ obtain ⟨e', he'⟩ := h₁,
rw ← he',
change _ = (ring.inverse ↑(e'.trans e.symm)) ∘L ↑e.symm,
ext,
simp },
{ suffices : ¬is_unit ((e.symm : M₂ →L[R] M).comp f),
{ simp [this, h₁] },
contrapose! h₁,
rcases h₁ with ⟨F, hF⟩,
use (contin... | lemma | continuous_linear_map.to_ring_inverse | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"coe_fn_coe_base'",
"continuous_linear_equiv.units_equiv",
"is_unit",
"ring.inverse"
] | The function `continuous_linear_equiv.inverse` can be written in terms of `ring.inverse` for the
ring of self-maps of the domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_inverse_eq_map_inverse : ring.inverse = @inverse R M M _ _ _ _ _ _ _ | begin
ext,
simp [to_ring_inverse (continuous_linear_equiv.refl R M)],
end | lemma | continuous_linear_map.ring_inverse_eq_map_inverse | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_linear_equiv.refl",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_complemented (p : submodule R M) : Prop | ∃ f : M →L[R] p, ∀ x : p, f x = x | def | submodule.closed_complemented | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule"
] | A submodule `p` is called *complemented* if there exists a continuous projection `M →ₗ[R] p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_complemented.has_closed_complement {p : submodule R M} [t1_space p]
(h : closed_complemented p) :
∃ (q : submodule R M) (hq : is_closed (q : set M)), is_compl p q | exists.elim h $ λ f hf, ⟨ker f, f.is_closed_ker, linear_map.is_compl_of_proj hf⟩ | lemma | submodule.closed_complemented.has_closed_complement | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"is_closed",
"is_compl",
"linear_map.is_compl_of_proj",
"submodule",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_complemented.is_closed [topological_add_group M] [t1_space M]
{p : submodule R M} (h : closed_complemented p) :
is_closed (p : set M) | begin
rcases h with ⟨f, hf⟩,
have : ker (id R M - p.subtypeL.comp f) = p := linear_map.ker_id_sub_eq_of_proj hf,
exact this ▸ (is_closed_ker _)
end | lemma | submodule.closed_complemented.is_closed | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"is_closed",
"linear_map.ker_id_sub_eq_of_proj",
"submodule",
"t1_space",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_complemented_bot : closed_complemented (⊥ : submodule R M) | ⟨0, λ x, by simp only [zero_apply, eq_zero_of_bot_submodule x]⟩ | lemma | submodule.closed_complemented_bot | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_complemented_top : closed_complemented (⊤ : submodule R M) | ⟨(id R M).cod_restrict ⊤ (λ x, trivial), λ x, subtype.ext_iff_val.2 $ by simp⟩ | lemma | submodule.closed_complemented_top | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.closed_complemented_ker_of_right_inverse {R : Type*} [ring R]
{M : Type*} [topological_space M] [add_comm_group M]
{M₂ : Type*} [topological_space M₂] [add_comm_group M₂] [module R M] [module R M₂]
[topological_add_group M] (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M)
(h : function.right_inverse f₂... | ⟨f₁.proj_ker_of_right_inverse f₂ h, f₁.proj_ker_of_right_inverse_apply_idem f₂ h⟩ | lemma | continuous_linear_map.closed_complemented_ker_of_right_inverse | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"add_comm_group",
"module",
"ring",
"topological_add_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_mkq [topological_add_group M] : is_open_map S.mkq | quotient_add_group.is_open_map_coe S.to_add_subgroup | lemma | submodule.is_open_map_mkq | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"is_open_map",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_add_group_quotient [topological_add_group M] :
topological_add_group (M ⧸ S) | topological_add_group_quotient S.to_add_subgroup | instance | submodule.topological_add_group_quotient | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_smul_quotient [topological_space R] [topological_add_group M]
[has_continuous_smul R M] :
has_continuous_smul R (M ⧸ S) | begin
split,
have quot : quotient_map (λ au : R × M, (au.1, S.mkq au.2)),
from is_open_map.to_quotient_map
(is_open_map.id.prod S.is_open_map_mkq)
(continuous_id.prod_map continuous_quot_mk)
(function.surjective_id.prod_map $ surjective_quot_mk _),
rw quot.continuous_iff,
exact continuous_... | instance | submodule.has_continuous_smul_quotient | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"continuous_quot_mk",
"has_continuous_smul",
"is_open_map.to_quotient_map",
"quotient_map",
"surjective_quot_mk",
"topological_add_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t3_quotient_of_is_closed [topological_add_group M] [is_closed (S : set M)] :
t3_space (M ⧸ S) | begin
letI : is_closed (S.to_add_subgroup : set M) := ‹_›,
exact S.to_add_subgroup.t3_quotient_of_is_closed
end | instance | submodule.t3_quotient_of_is_closed | topology.algebra.module | src/topology/algebra/module/basic.lean | [
"topology.algebra.ring.basic",
"topology.algebra.mul_action",
"topology.algebra.uniform_group",
"topology.continuous_function.basic",
"topology.uniform_space.uniform_embedding",
"algebra.algebra.basic",
"linear_algebra.projection",
"linear_algebra.pi"
] | [
"is_closed",
"t3_space",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
character_space (𝕜 : Type*) (A : Type*) [comm_semiring 𝕜] [topological_space 𝕜]
[has_continuous_add 𝕜] [has_continuous_const_smul 𝕜 𝕜]
[non_unital_non_assoc_semiring A] [topological_space A] [module 𝕜 A] | {φ : weak_dual 𝕜 A | (φ ≠ 0) ∧ (∀ (x y : A), φ (x * y) = (φ x) * (φ y))} | def | weak_dual.character_space | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"comm_semiring",
"has_continuous_add",
"has_continuous_const_smul",
"module",
"non_unital_non_assoc_semiring",
"topological_space",
"weak_dual"
] | The character space of a topological algebra is the subset of elements of the weak dual that
are also algebra homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_coe (φ : character_space 𝕜 A) : ⇑(φ : weak_dual 𝕜 A) = φ | rfl | lemma | weak_dual.character_space.coe_coe | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"coe_coe",
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {φ ψ : character_space 𝕜 A} (h : ∀ x, φ x = ψ x) : φ = ψ | fun_like.ext _ _ h | lemma | weak_dual.character_space.ext | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_clm (φ : character_space 𝕜 A) : A →L[𝕜] 𝕜 | (φ : weak_dual 𝕜 A) | def | weak_dual.character_space.to_clm | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"weak_dual"
] | An element of the character space, as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_clm (φ : character_space 𝕜 A) : ⇑(to_clm φ) = φ | rfl | lemma | weak_dual.character_space.coe_to_clm | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_alg_hom (φ : character_space 𝕜 A) : A →ₙₐ[𝕜] 𝕜 | { to_fun := (φ : A → 𝕜),
map_mul' := map_mul φ,
map_smul' := map_smul φ,
map_zero' := map_zero φ,
map_add' := map_add φ } | def | weak_dual.character_space.to_non_unital_alg_hom | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"map_mul"
] | An element of the character space, as an non-unital algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_non_unital_alg_hom (φ : character_space 𝕜 A) : ⇑(to_non_unital_alg_hom φ) = φ | rfl | lemma | weak_dual.character_space.coe_to_non_unital_alg_hom | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
union_zero :
character_space 𝕜 A ∪ {0} = {φ : weak_dual 𝕜 A | ∀ (x y : A), φ (x * y) = (φ x) * (φ y)} | le_antisymm
(by { rintros φ (hφ | h₀), { exact hφ.2 }, { exact λ x y, by simp [set.eq_of_mem_singleton h₀] }})
(λ φ hφ, or.elim (em $ φ = 0) (λ h₀, or.inr h₀) (λ h₀, or.inl ⟨h₀, hφ⟩)) | lemma | weak_dual.character_space.union_zero | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"em",
"set.eq_of_mem_singleton",
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
union_zero_is_closed [t2_space 𝕜] [has_continuous_mul 𝕜] :
is_closed (character_space 𝕜 A ∪ {0}) | begin
simp only [union_zero, set.set_of_forall],
exact is_closed_Inter (λ x, is_closed_Inter $ λ y, is_closed_eq (eval_continuous _) $
(eval_continuous _).mul (eval_continuous _))
end | lemma | weak_dual.character_space.union_zero_is_closed | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"has_continuous_mul",
"is_closed",
"is_closed_Inter",
"is_closed_eq",
"set.set_of_forall",
"t2_space"
] | The `character_space 𝕜 A` along with `0` is always a closed set in `weak_dual 𝕜 A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_alg_hom (φ : character_space 𝕜 A) : A →ₐ[𝕜] 𝕜 | { map_one' := map_one φ,
commutes' := alg_hom_class.commutes φ,
..to_non_unital_alg_hom φ } | def | weak_dual.character_space.to_alg_hom | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"map_one"
] | An element of the character space of a unital algebra, as an algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_set_map_one_map_mul [nontrivial 𝕜] : character_space 𝕜 A =
{φ : weak_dual 𝕜 A | (φ 1 = 1) ∧ (∀ (x y : A), φ (x * y) = (φ x) * (φ y))} | begin
ext x,
refine ⟨λ h, ⟨map_one (⟨x, h⟩ : character_space 𝕜 A), h.2⟩, λ h, ⟨_, h.2⟩⟩,
rintro rfl,
simpa using h.1,
end | lemma | weak_dual.character_space.eq_set_map_one_map_mul | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"nontrivial",
"weak_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed [nontrivial 𝕜] [t2_space 𝕜] [has_continuous_mul 𝕜] :
is_closed (character_space 𝕜 A) | begin
rw [eq_set_map_one_map_mul, set.set_of_and],
refine is_closed.inter (is_closed_eq (eval_continuous _) continuous_const) _,
simpa only [(union_zero 𝕜 A).symm] using union_zero_is_closed _ _,
end | lemma | weak_dual.character_space.is_closed | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"continuous_const",
"has_continuous_mul",
"is_closed",
"is_closed.inter",
"is_closed_eq",
"nontrivial",
"set.set_of_and",
"t2_space"
] | under suitable mild assumptions on `𝕜`, the character space is a closed set in
`weak_dual 𝕜 A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_mem_spectrum [nontrivial 𝕜] (φ : character_space 𝕜 A) (a : A) : φ a ∈ spectrum 𝕜 a | alg_hom.apply_mem_spectrum φ a | lemma | weak_dual.character_space.apply_mem_spectrum | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"alg_hom.apply_mem_spectrum",
"nontrivial",
"spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_ker {φ ψ : character_space 𝕜 A} (h : ring_hom.ker φ = ring_hom.ker ψ) : φ = ψ | begin
ext,
have : x - algebra_map 𝕜 A (ψ x) ∈ ring_hom.ker φ,
{ simpa only [h, ring_hom.mem_ker, map_sub, alg_hom_class.commutes] using sub_self (ψ x) },
{ rwa [ring_hom.mem_ker, map_sub, alg_hom_class.commutes, sub_eq_zero] at this, }
end | lemma | weak_dual.character_space.ext_ker | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"algebra_map",
"ring_hom.ker",
"ring_hom.mem_ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_is_maximal (φ : character_space 𝕜 A) : (ring_hom.ker φ).is_maximal | ring_hom.ker_is_maximal_of_surjective φ $ λ z, ⟨algebra_map 𝕜 A z,
by simp only [alg_hom_class.commutes, algebra.id.map_eq_id, ring_hom.id_apply]⟩ | instance | weak_dual.ker_is_maximal | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"algebra.id.map_eq_id",
"ring_hom.id_apply",
"ring_hom.ker",
"ring_hom.ker_is_maximal_of_surjective"
] | The `ring_hom.ker` of `φ : character_space 𝕜 A` is maximal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gelfand_transform : A →ₐ[𝕜] C(character_space 𝕜 A, 𝕜) | { to_fun := λ a,
{ to_fun := λ φ, φ a,
continuous_to_fun := (eval_continuous a).comp continuous_induced_dom },
map_one' := by {ext, simp only [coe_mk, coe_one, pi.one_apply, map_one a] },
map_mul' := λ a b, by {ext, simp only [map_mul, coe_mk, coe_mul, pi.mul_apply] },
map_zero' := by {ext, simp only ... | def | weak_dual.gelfand_transform | topology.algebra.module | src/topology/algebra/module/character_space.lean | [
"topology.algebra.module.weak_dual",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"algebra.id.map_eq_id",
"algebra.id.smul_eq_mul",
"algebra_map_apply",
"continuous_induced_dom",
"map_mul",
"map_one",
"mul_one",
"pi.mul_apply",
"pi.one_apply",
"ring_hom.id_apply"
] | The **Gelfand transform** is an algebra homomorphism (over `𝕜`) from a topological `𝕜`-algebra
`A` into the `𝕜`-algebra of continuous `𝕜`-valued functions on the `character_space 𝕜 A`.
The character space itself consists of all algebra homomorphisms from `A` to `𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
det {R : Type*} [comm_ring R]
{M : Type*} [topological_space M] [add_comm_group M] [module R M] (A : M →L[R] M) : R | linear_map.det (A : M →ₗ[R] M) | def | continuous_linear_map.det | topology.algebra.module | src/topology/algebra/module/determinant.lean | [
"topology.algebra.module.basic",
"linear_algebra.determinant"
] | [
"add_comm_group",
"comm_ring",
"linear_map.det",
"module",
"topological_space"
] | The determinant of a continuous linear map, mainly as a convenience device to be able to
write `A.det` instead of `(A : M →ₗ[R] M).det`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
det_coe_symm {R : Type*} [field R]
{M : Type*} [topological_space M] [add_comm_group M] [module R M] (A : M ≃L[R] M) :
(A.symm : M →L[R] M).det = (A : M →L[R] M).det ⁻¹ | linear_equiv.det_coe_symm A.to_linear_equiv | lemma | continuous_linear_equiv.det_coe_symm | topology.algebra.module | src/topology/algebra/module/determinant.lean | [
"topology.algebra.module.basic",
"linear_algebra.determinant"
] | [
"add_comm_group",
"field",
"linear_equiv.det_coe_symm",
"module",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_topology_of_t2 {t : topological_space 𝕜}
(h₁ : @topological_add_group 𝕜 t _)
(h₂ : @has_continuous_smul 𝕜 𝕜 _ hnorm.to_uniform_space.to_topological_space t)
(h₃ : @t2_space 𝕜 t) :
t = hnorm.to_uniform_space.to_topological_space | begin
-- Let `𝓣₀` denote the topology on `𝕜` induced by the norm, and `𝓣` be any T2 vector
-- topology on `𝕜`. To show that `𝓣₀ = 𝓣`, it suffices to show that they have the same
-- neighborhoods of 0.
refine topological_add_group.ext h₁ infer_instance (le_antisymm _ _),
{ -- To show `𝓣 ≤ 𝓣₀`, we have ... | lemma | unique_topology_of_t2 | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"balanced_core",
"balanced_core_balanced",
"balanced_core_mem_nhds_zero",
"balanced_core_subset",
"has_continuous_smul",
"inv_mul_cancel",
"is_open.mem_nhds",
"is_open_compl_singleton",
"metric.mem_closed_ball_self",
"mul_assoc",
"mul_inv_le_iff",
"mul_one",
"nhds",
"norm_inv",
"norm_mul... | If `𝕜` is a nontrivially normed field, any T2 topology on `𝕜` which makes it a topological
vector space over itself (with the norm topology) is *equal* to the norm topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.continuous_of_is_closed_ker (l : E →ₗ[𝕜] 𝕜) (hl : is_closed (l.ker : set E)) :
continuous l | begin
-- `l` is either constant or surjective. If it is constant, the result is trivial.
by_cases H : finrank 𝕜 l.range = 0,
{ rw [finrank_eq_zero, linear_map.range_eq_bot] at H,
rw H,
exact continuous_zero },
{ -- In the case where `l` is surjective, we factor it as `φ : (E ⧸ l.ker) ≃ₗ[𝕜] 𝕜`. Note t... | lemma | linear_map.continuous_of_is_closed_ker | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"continuous",
"continuous_coinduced_rng",
"continuous_induced_dom",
"continuous_quot_mk",
"equiv.induced_symm",
"finrank_eq_zero",
"has_continuous_smul_induced",
"is_closed",
"linear_equiv.of_bijective",
"linear_map.ker_eq_bot",
"linear_map.range_eq_bot",
"linear_map.range_eq_top",
"separate... | Any linear form on a topological vector space over a nontrivially normed field is continuous if
its kernel is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.continuous_iff_is_closed_ker (l : E →ₗ[𝕜] 𝕜) :
continuous l ↔ is_closed (l.ker : set E) | ⟨λ h, is_closed_singleton.preimage h, l.continuous_of_is_closed_ker⟩ | lemma | linear_map.continuous_iff_is_closed_ker | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"continuous",
"is_closed"
] | Any linear form on a topological vector space over a nontrivially normed field is continuous if
and only if its kernel is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.continuous_of_nonzero_on_open (l : E →ₗ[𝕜] 𝕜) (s : set E) (hs₁ : is_open s)
(hs₂ : s.nonempty) (hs₃ : ∀ x ∈ s, l x ≠ 0) : continuous l | begin
refine l.continuous_of_is_closed_ker (l.is_closed_or_dense_ker.resolve_right $ λ hl, _),
rcases hs₂ with ⟨x, hx⟩,
have : x ∈ interior (l.ker : set E)ᶜ,
{ rw mem_interior_iff_mem_nhds,
exact mem_of_superset (hs₁.mem_nhds hx) hs₃ },
rwa hl.interior_compl at this
end | lemma | linear_map.continuous_of_nonzero_on_open | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"continuous",
"interior",
"is_open",
"mem_interior_iff_mem_nhds"
] | Over a nontrivially normed field, any linear form which is nonzero on a nonempty open set is
automatically continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_equiv_fun_basis_aux [ht2 : t2_space E] {ι : Type v} [fintype ι]
(ξ : basis ι 𝕜 E) : continuous ξ.equiv_fun | begin
letI : uniform_space E := topological_add_group.to_uniform_space E,
letI : uniform_add_group E := topological_add_comm_group_is_uniform,
letI : separated_space E := separated_iff_t2.mpr ht2,
unfreezingI { induction hn : fintype.card ι with n IH generalizing ι E },
{ rw fintype.card_eq_zero_iff at hn,
... | lemma | continuous_equiv_fun_basis_aux | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"basis",
"basis.of_vector_space",
"basis.of_vector_space_index",
"complete_space_congr",
"continuous",
"continuous_of_const",
"continuous_pi_iff",
"finite_dimensional",
"finrank_eq_zero",
"fintype",
"fintype.card",
"fintype.card_eq_zero_iff",
"is_closed",
"is_complete",
"linear_map.conti... | This version imposes `ι` and `E` to live in the same universe, so you should instead use
`continuous_equiv_fun_basis` which gives the same result without universe restrictions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.continuous_of_finite_dimensional [t2_space E] [finite_dimensional 𝕜 E]
(f : E →ₗ[𝕜] F') :
continuous f | begin
-- for the proof, go to a model vector space `b → 𝕜` thanks to `continuous_equiv_fun_basis`, and
-- argue that all linear maps there are continuous.
let b := basis.of_vector_space 𝕜 E,
have A : continuous b.equiv_fun :=
continuous_equiv_fun_basis_aux b,
have B : continuous (f.comp (b.equiv_fun.sym... | theorem | linear_map.continuous_of_finite_dimensional | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"basis.equiv_fun_symm_apply",
"basis.of_vector_space",
"basis.of_vector_space_index",
"basis.sum_repr",
"continuous",
"continuous_equiv_fun_basis_aux",
"finite_dimensional",
"linear_map.continuous_on_pi",
"t2_space"
] | Any linear map on a finite dimensional space over a complete field is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.continuous_linear_map_class_of_finite_dimensional
[t2_space E] [finite_dimensional 𝕜 E] :
continuous_linear_map_class (E →ₗ[𝕜] F') 𝕜 E F' | { map_continuous := λ f, f.continuous_of_finite_dimensional,
..linear_map.semilinear_map_class } | instance | linear_map.continuous_linear_map_class_of_finite_dimensional | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"continuous_linear_map_class",
"finite_dimensional",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_equiv_fun_basis [t2_space E] {ι : Type*} [fintype ι] (ξ : basis ι 𝕜 E) :
continuous ξ.equiv_fun | begin
haveI : finite_dimensional 𝕜 E := of_fintype_basis ξ,
exact ξ.equiv_fun.to_linear_map.continuous_of_finite_dimensional
end | theorem | continuous_equiv_fun_basis | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"basis",
"continuous",
"finite_dimensional",
"fintype",
"t2_space"
] | In finite dimensions over a non-discrete complete normed field, the canonical identification
(in terms of a basis) with `𝕜^n` (endowed with the product topology) is continuous.
This is the key fact wich makes all linear maps from a T2 finite dimensional TVS over such a field
continuous (see `linear_map.continuous_of_f... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_linear_map : (E →ₗ[𝕜] F') ≃ₗ[𝕜] E →L[𝕜] F' | { to_fun := λ f, ⟨f, f.continuous_of_finite_dimensional⟩,
inv_fun := coe,
map_add' := λ f g, rfl,
map_smul' := λ c f, rfl,
left_inv := λ f, rfl,
right_inv := λ f, continuous_linear_map.coe_injective rfl } | def | linear_map.to_continuous_linear_map | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"continuous_linear_map.coe_injective",
"inv_fun"
] | The continuous linear map induced by a linear map on a finite dimensional space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_continuous_linear_map' (f : E →ₗ[𝕜] F') :
⇑f.to_continuous_linear_map = f | rfl | lemma | linear_map.coe_to_continuous_linear_map' | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_continuous_linear_map (f : E →ₗ[𝕜] F') :
(f.to_continuous_linear_map : E →ₗ[𝕜] F') = f | rfl | lemma | linear_map.coe_to_continuous_linear_map | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_continuous_linear_map_symm :
⇑(to_continuous_linear_map : (E →ₗ[𝕜] F') ≃ₗ[𝕜] E →L[𝕜] F').symm = coe | rfl | lemma | linear_map.coe_to_continuous_linear_map_symm | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
det_to_continuous_linear_map (f : E →ₗ[𝕜] E) :
f.to_continuous_linear_map.det = f.det | rfl | lemma | linear_map.det_to_continuous_linear_map | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_to_continuous_linear_map (f : E →ₗ[𝕜] F') :
ker f.to_continuous_linear_map = ker f | rfl | lemma | linear_map.ker_to_continuous_linear_map | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_to_continuous_linear_map (f : E →ₗ[𝕜] F') :
range f.to_continuous_linear_map = range f | rfl | lemma | linear_map.range_to_continuous_linear_map | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_of_finite_dimensional (f : F →ₗ[𝕜] E) (hf : function.surjective f) :
is_open_map f | begin
rcases f.exists_right_inverse_of_surjective (linear_map.range_eq_top.2 hf) with ⟨g, hg⟩,
refine is_open_map.of_sections (λ x, ⟨λ y, g (y - f x) + x, _, _, λ y, _⟩),
{ exact ((g.continuous_of_finite_dimensional.comp $ continuous_id.sub continuous_const).add
continuous_const).continuous_at },
{ rw [su... | lemma | linear_map.is_open_map_of_finite_dimensional | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"continuous_at",
"continuous_const",
"is_open_map",
"is_open_map.of_sections"
] | A surjective linear map `f` with finite dimensional codomain is an open map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
can_lift_continuous_linear_map : can_lift (E →ₗ[𝕜] F) (E →L[𝕜] F) coe (λ _, true) | ⟨λ f _, ⟨f.to_continuous_linear_map, rfl⟩⟩ | instance | linear_map.can_lift_continuous_linear_map | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"can_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_linear_equiv (e : E ≃ₗ[𝕜] F) : E ≃L[𝕜] F | { continuous_to_fun := e.to_linear_map.continuous_of_finite_dimensional,
continuous_inv_fun := begin
haveI : finite_dimensional 𝕜 F := e.finite_dimensional,
exact e.symm.to_linear_map.continuous_of_finite_dimensional
end,
..e } | def | linear_equiv.to_continuous_linear_equiv | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"finite_dimensional"
] | The continuous linear equivalence induced by a linear equivalence on a finite dimensional
space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_continuous_linear_equiv (e : E ≃ₗ[𝕜] F) :
(e.to_continuous_linear_equiv : E →ₗ[𝕜] F) = e | rfl | lemma | linear_equiv.coe_to_continuous_linear_equiv | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_continuous_linear_equiv' (e : E ≃ₗ[𝕜] F) :
(e.to_continuous_linear_equiv : E → F) = e | rfl | lemma | linear_equiv.coe_to_continuous_linear_equiv' | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_continuous_linear_equiv_symm (e : E ≃ₗ[𝕜] F) :
(e.to_continuous_linear_equiv.symm : F →ₗ[𝕜] E) = e.symm | rfl | lemma | linear_equiv.coe_to_continuous_linear_equiv_symm | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_continuous_linear_equiv_symm' (e : E ≃ₗ[𝕜] F) :
(e.to_continuous_linear_equiv.symm : F → E) = e.symm | rfl | lemma | linear_equiv.coe_to_continuous_linear_equiv_symm' | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv_to_continuous_linear_equiv (e : E ≃ₗ[𝕜] F) :
e.to_continuous_linear_equiv.to_linear_equiv = e | by { ext x, refl } | lemma | linear_equiv.to_linear_equiv_to_continuous_linear_equiv | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv_to_continuous_linear_equiv_symm (e : E ≃ₗ[𝕜] F) :
e.to_continuous_linear_equiv.symm.to_linear_equiv = e.symm | by { ext x, refl } | lemma | linear_equiv.to_linear_equiv_to_continuous_linear_equiv_symm | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
can_lift_continuous_linear_equiv :
can_lift (E ≃ₗ[𝕜] F) (E ≃L[𝕜] F) continuous_linear_equiv.to_linear_equiv (λ _, true) | ⟨λ f _, ⟨_, f.to_linear_equiv_to_continuous_linear_equiv⟩⟩ | instance | linear_equiv.can_lift_continuous_linear_equiv | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"can_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_dimensional.nonempty_continuous_linear_equiv_of_finrank_eq
(cond : finrank 𝕜 E = finrank 𝕜 F) : nonempty (E ≃L[𝕜] F) | (nonempty_linear_equiv_of_finrank_eq cond).map linear_equiv.to_continuous_linear_equiv | theorem | finite_dimensional.nonempty_continuous_linear_equiv_of_finrank_eq | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"linear_equiv.to_continuous_linear_equiv"
] | Two finite-dimensional topological vector spaces over a complete normed field are continuously
linearly equivalent if they have the same (finite) dimension. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_dimensional.nonempty_continuous_linear_equiv_iff_finrank_eq :
nonempty (E ≃L[𝕜] F) ↔ finrank 𝕜 E = finrank 𝕜 F | ⟨ λ ⟨h⟩, h.to_linear_equiv.finrank_eq,
λ h, finite_dimensional.nonempty_continuous_linear_equiv_of_finrank_eq h ⟩ | theorem | finite_dimensional.nonempty_continuous_linear_equiv_iff_finrank_eq | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"finite_dimensional.nonempty_continuous_linear_equiv_of_finrank_eq"
] | Two finite-dimensional topological vector spaces over a complete normed field are continuously
linearly equivalent if and only if they have the same (finite) dimension. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.of_finrank_eq
(cond : finrank 𝕜 E = finrank 𝕜 F) : E ≃L[𝕜] F | (linear_equiv.of_finrank_eq E F cond).to_continuous_linear_equiv | def | continuous_linear_equiv.of_finrank_eq | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"linear_equiv.of_finrank_eq"
] | A continuous linear equivalence between two finite-dimensional topological vector spaces over a
complete normed field of the same (finite) dimension. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
constrL (v : basis ι 𝕜 E) (f : ι → F) :
E →L[𝕜] F | by haveI : finite_dimensional 𝕜 E := finite_dimensional.of_fintype_basis v;
exact (v.constr 𝕜 f).to_continuous_linear_map | def | basis.constrL | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"basis",
"finite_dimensional",
"finite_dimensional.of_fintype_basis"
] | Construct a continuous linear map given the value at a finite basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_constrL (v : basis ι 𝕜 E) (f : ι → F) :
(v.constrL f : E →ₗ[𝕜] F) = v.constr 𝕜 f | rfl | lemma | basis.coe_constrL | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_funL (v : basis ι 𝕜 E) : E ≃L[𝕜] (ι → 𝕜) | { continuous_to_fun := begin
haveI : finite_dimensional 𝕜 E := finite_dimensional.of_fintype_basis v,
exact v.equiv_fun.to_linear_map.continuous_of_finite_dimensional,
end,
continuous_inv_fun := begin
change continuous v.equiv_fun.symm.to_fun,
exact v.equiv_fun.symm.to_linear_map.continuous_of_fini... | def | basis.equiv_funL | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"basis",
"continuous",
"finite_dimensional",
"finite_dimensional.of_fintype_basis"
] | The continuous linear equivalence between a vector space over `𝕜` with a finite basis and
functions from its basis indexing type to `𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
constrL_apply (v : basis ι 𝕜 E) (f : ι → F) (e : E) :
(v.constrL f) e = ∑ i, (v.equiv_fun e i) • f i | v.constr_apply_fintype 𝕜 _ _ | lemma | basis.constrL_apply | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constrL_basis (v : basis ι 𝕜 E) (f : ι → F) (i : ι) :
(v.constrL f) (v i) = f i | v.constr_basis 𝕜 _ _ | lemma | basis.constrL_basis | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_linear_equiv_of_det_ne_zero
(f : E →L[𝕜] E) (hf : f.det ≠ 0) : E ≃L[𝕜] E | ((f : E →ₗ[𝕜] E).equiv_of_det_ne_zero hf).to_continuous_linear_equiv | def | continuous_linear_map.to_continuous_linear_equiv_of_det_ne_zero | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | Builds a continuous linear equivalence from a continuous linear map on a finite-dimensional
vector space whose determinant is nonzero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_continuous_linear_equiv_of_det_ne_zero (f : E →L[𝕜] E) (hf : f.det ≠ 0) :
(f.to_continuous_linear_equiv_of_det_ne_zero hf : E →L[𝕜] E) = f | by { ext x, refl } | lemma | continuous_linear_map.coe_to_continuous_linear_equiv_of_det_ne_zero | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_linear_equiv_of_det_ne_zero_apply
(f : E →L[𝕜] E) (hf : f.det ≠ 0) (x : E) :
f.to_continuous_linear_equiv_of_det_ne_zero hf x = f x | rfl | lemma | continuous_linear_map.to_continuous_linear_equiv_of_det_ne_zero_apply | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.matrix.to_lin_fin_two_prod_to_continuous_linear_map (a b c d : 𝕜) :
(matrix.to_lin (basis.fin_two_prod 𝕜) (basis.fin_two_prod 𝕜)
!![a, b; c, d]).to_continuous_linear_map =
(a • continuous_linear_map.fst 𝕜 𝕜 𝕜 + b • continuous_linear_map.snd 𝕜 𝕜 𝕜).prod
(c • continuous_linear_map.fst 𝕜 𝕜 𝕜... | continuous_linear_map.ext $ matrix.to_lin_fin_two_prod_apply _ _ _ _ | lemma | matrix.to_lin_fin_two_prod_to_continuous_linear_map | topology.algebra.module | src/topology/algebra/module/finite_dimension.lean | [
"analysis.locally_convex.balanced_core_hull",
"linear_algebra.free_module.finite.matrix",
"topology.algebra.module.simple",
"topology.algebra.module.determinant"
] | [
"basis.fin_two_prod",
"continuous_linear_map.ext",
"continuous_linear_map.fst",
"continuous_linear_map.snd",
"matrix.to_lin",
"matrix.to_lin_fin_two_prod_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed (f : E →ₗ.[R] F) : Prop | is_closed (f.graph : set (E × F)) | def | linear_pmap.is_closed | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [
"is_closed"
] | An unbounded operator is closed iff its graph is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closable (f : E →ₗ.[R] F) : Prop | ∃ (f' : linear_pmap R E F), f.graph.topological_closure = f'.graph | def | linear_pmap.is_closable | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [
"linear_pmap"
] | An unbounded operator is closable iff the closure of its graph is a graph. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed.is_closable {f : E →ₗ.[R] F} (hf : f.is_closed) : f.is_closable | ⟨f, hf.submodule_topological_closure_eq⟩ | lemma | linear_pmap.is_closed.is_closable | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [] | A closed operator is trivially closable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closable.le_is_closable {f g : E →ₗ.[R] F} (hf : f.is_closable) (hfg : g ≤ f) :
g.is_closable | begin
cases hf with f' hf,
have : g.graph.topological_closure ≤ f'.graph :=
by { rw ←hf, exact submodule.topological_closure_mono (le_graph_of_le hfg) },
refine ⟨g.graph.topological_closure.to_linear_pmap _, _⟩,
{ intros x hx hx',
cases x,
exact f'.graph_fst_eq_zero_snd (this hx) hx' },
rw [submodul... | lemma | linear_pmap.is_closable.le_is_closable | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [
"submodule.to_linear_pmap_graph_eq",
"submodule.topological_closure_mono"
] | If `g` has a closable extension `f`, then `g` itself is closable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closable.exists_unique {f : E →ₗ.[R] F} (hf : f.is_closable) :
∃! (f' : E →ₗ.[R] F), f.graph.topological_closure = f'.graph | begin
refine exists_unique_of_exists_of_unique hf (λ _ _ hy₁ hy₂, eq_of_eq_graph _),
rw [←hy₁, ←hy₂],
end | lemma | linear_pmap.is_closable.exists_unique | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [] | The closure is unique. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure (f : E →ₗ.[R] F) : E →ₗ.[R] F | if hf : f.is_closable then hf.some else f | def | linear_pmap.closure | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [
"closure"
] | If `f` is closable, then `f.closure` is the closure. Otherwise it is defined
as `f.closure = f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_def {f : E →ₗ.[R] F} (hf : f.is_closable) :
f.closure = hf.some | by simp [closure, hf] | lemma | linear_pmap.closure_def | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_def' {f : E →ₗ.[R] F} (hf : ¬f.is_closable) :
f.closure = f | by simp [closure, hf] | lemma | linear_pmap.closure_def' | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closable.graph_closure_eq_closure_graph {f : E →ₗ.[R] F} (hf : f.is_closable) :
f.graph.topological_closure = f.closure.graph | begin
rw closure_def hf,
exact hf.some_spec,
end | lemma | linear_pmap.is_closable.graph_closure_eq_closure_graph | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [] | The closure (as a submodule) of the graph is equal to the graph of the closure
(as a `linear_pmap`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_closure (f : E →ₗ.[R] F) : f ≤ f.closure | begin
by_cases hf : f.is_closable,
{ refine le_of_le_graph _,
rw ←hf.graph_closure_eq_closure_graph,
exact (graph f).le_topological_closure },
rw closure_def' hf,
end | lemma | linear_pmap.le_closure | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [] | A `linear_pmap` is contained in its closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closable.closure_mono {f g : E →ₗ.[R] F} (hg : g.is_closable) (h : f ≤ g) :
f.closure ≤ g.closure | begin
refine le_of_le_graph _,
rw ←(hg.le_is_closable h).graph_closure_eq_closure_graph,
rw ←hg.graph_closure_eq_closure_graph,
exact submodule.topological_closure_mono (le_graph_of_le h),
end | lemma | linear_pmap.is_closable.closure_mono | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [
"submodule.topological_closure_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closable.closure_is_closed {f : E →ₗ.[R] F} (hf : f.is_closable) :
f.closure.is_closed | begin
rw [is_closed, ←hf.graph_closure_eq_closure_graph],
exact f.graph.is_closed_topological_closure,
end | lemma | linear_pmap.is_closable.closure_is_closed | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [
"is_closed"
] | If `f` is closable, then the closure is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closable.closure_is_closable {f : E →ₗ.[R] F} (hf : f.is_closable) :
f.closure.is_closable | hf.closure_is_closed.is_closable | lemma | linear_pmap.is_closable.closure_is_closable | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [] | If `f` is closable, then the closure is closable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closable_iff_exists_closed_extension {f : E →ₗ.[R] F} : f.is_closable ↔
∃ (g : E →ₗ.[R] F) (hg : g.is_closed), f ≤ g | ⟨λ h, ⟨f.closure, h.closure_is_closed, f.le_closure⟩, λ ⟨_, hg, h⟩, hg.is_closable.le_is_closable h⟩ | lemma | linear_pmap.is_closable_iff_exists_closed_extension | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_core (f : E →ₗ.[R] F) (S : submodule R E) : Prop | (le_domain : S ≤ f.domain)
(closure_eq : (f.dom_restrict S).closure = f) | structure | linear_pmap.has_core | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [
"closure",
"submodule"
] | A submodule `S` is a core of `f` if the closure of the restriction of `f` to `S` is again `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_core_def {f : E →ₗ.[R] F} {S : submodule R E} (h : f.has_core S) :
(f.dom_restrict S).closure = f | h.2 | lemma | linear_pmap.has_core_def | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [
"closure",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_has_core (f : E →ₗ.[R] F) : f.closure.has_core f.domain | begin
refine ⟨f.le_closure.1, _⟩,
congr,
ext,
{ simp only [dom_restrict_domain, submodule.mem_inf, and_iff_left_iff_imp],
intro hx,
exact f.le_closure.1 hx },
intros x y hxy,
let z : f.closure.domain := ⟨y.1, f.le_closure.1 y.2⟩,
have hyz : (y : E) = z := by simp,
rw f.le_closure.2 hyz,
exact ... | lemma | linear_pmap.closure_has_core | topology.algebra.module | src/topology/algebra/module/linear_pmap.lean | [
"linear_algebra.linear_pmap",
"topology.algebra.module.basic"
] | [
"and_iff_left_iff_imp",
"submodule.mem_inf"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_convex_space (𝕜 E : Type*) [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E]
[topological_space E] : Prop | (convex_basis : ∀ x : E, (𝓝 x).has_basis (λ (s : set E), s ∈ 𝓝 x ∧ convex 𝕜 s) id) | class | locally_convex_space | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"add_comm_monoid",
"convex",
"module",
"ordered_semiring",
"topological_space"
] | A `locally_convex_space` is a topological semimodule over an ordered semiring in which convex
neighborhoods of a point form a neighborhood basis at that point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_convex_space_iff :
locally_convex_space 𝕜 E ↔
∀ x : E, (𝓝 x).has_basis (λ (s : set E), s ∈ 𝓝 x ∧ convex 𝕜 s) id | ⟨@locally_convex_space.convex_basis _ _ _ _ _ _, locally_convex_space.mk⟩ | lemma | locally_convex_space_iff | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"convex",
"locally_convex_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_convex_space.of_bases {ι : Type*} (b : E → ι → set E) (p : E → ι → Prop)
(hbasis : ∀ x : E, (𝓝 x).has_basis (p x) (b x)) (hconvex : ∀ x i, p x i → convex 𝕜 (b x i)) :
locally_convex_space 𝕜 E | ⟨λ x, (hbasis x).to_has_basis
(λ i hi, ⟨b x i, ⟨⟨(hbasis x).mem_of_mem hi, hconvex x i hi⟩, le_refl (b x i)⟩⟩)
(λ s hs, ⟨(hbasis x).index s hs.1,
⟨(hbasis x).property_index hs.1, (hbasis x).set_index_subset hs.1⟩⟩)⟩ | lemma | locally_convex_space.of_bases | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"convex",
"locally_convex_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_convex_space.convex_basis_zero [locally_convex_space 𝕜 E] :
(𝓝 0 : filter E).has_basis (λ s, s ∈ (𝓝 0 : filter E) ∧ convex 𝕜 s) id | locally_convex_space.convex_basis 0 | lemma | locally_convex_space.convex_basis_zero | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"convex",
"filter",
"locally_convex_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_convex_space_iff_exists_convex_subset :
locally_convex_space 𝕜 E ↔ ∀ x : E, ∀ U ∈ 𝓝 x, ∃ S ∈ 𝓝 x, convex 𝕜 S ∧ S ⊆ U | (locally_convex_space_iff 𝕜 E).trans (forall_congr $ λ x, has_basis_self) | lemma | locally_convex_space_iff_exists_convex_subset | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"convex",
"locally_convex_space",
"locally_convex_space_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_convex_space.of_basis_zero {ι : Type*} (b : ι → set E) (p : ι → Prop)
(hbasis : (𝓝 0).has_basis p b) (hconvex : ∀ i, p i → convex 𝕜 (b i)) :
locally_convex_space 𝕜 E | begin
refine locally_convex_space.of_bases 𝕜 E (λ (x : E) (i : ι), ((+) x) '' b i) (λ _, p) (λ x, _)
(λ x i hi, (hconvex i hi).translate x),
rw ← map_add_left_nhds_zero,
exact hbasis.map _
end | lemma | locally_convex_space.of_basis_zero | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"convex",
"locally_convex_space",
"locally_convex_space.of_bases"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_convex_space_iff_zero :
locally_convex_space 𝕜 E ↔
(𝓝 0 : filter E).has_basis (λ (s : set E), s ∈ (𝓝 0 : filter E) ∧ convex 𝕜 s) id | ⟨λ h, @locally_convex_space.convex_basis _ _ _ _ _ _ h 0,
λ h, locally_convex_space.of_basis_zero 𝕜 E _ _ h (λ s, and.right)⟩ | lemma | locally_convex_space_iff_zero | topology.algebra.module | src/topology/algebra/module/locally_convex.lean | [
"analysis.convex.topology"
] | [
"convex",
"filter",
"locally_convex_space",
"locally_convex_space.of_basis_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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