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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|---|---|---|---|---|---|---|---|---|---|---|
norm_le_of_nonempty [nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M | @bounded_continuous_function.norm_le_of_nonempty _ _ _ _ _ (mk_of_compact f) _ | lemma | continuous_map.norm_le_of_nonempty | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"bounded_continuous_function.norm_le_of_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M | @bounded_continuous_function.norm_lt_iff_of_compact _ _ _ _ _ (mk_of_compact f) _ M0 | lemma | continuous_map.norm_lt_iff | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"bounded_continuous_function.norm_lt_iff_of_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ (x : α), ‖f x‖₊ < M | f.norm_lt_iff M0 | theorem | continuous_map.nnnorm_lt_iff | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_lt_iff_of_nonempty [nonempty α] {M : ℝ} :
‖f‖ < M ↔ ∀ x, ‖f x‖ < M | @bounded_continuous_function.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mk_of_compact f) _ | lemma | continuous_map.norm_lt_iff_of_nonempty | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"bounded_continuous_function.norm_lt_iff_of_nonempty_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_lt_iff_of_nonempty [nonempty α] {M : ℝ≥0} :
‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M | f.norm_lt_iff_of_nonempty | lemma | continuous_map.nnnorm_lt_iff_of_nonempty | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ | le_trans (le_abs.mpr (or.inl (le_refl (f x)))) (f.norm_coe_le_norm x) | lemma | continuous_map.apply_le_norm | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x | le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x))) | lemma | continuous_map.neg_norm_le_apply | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"neg_le_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_supr_norm : ‖f‖ = ⨆ x : α, ‖f x‖ | (mk_of_compact f).norm_eq_supr_norm | lemma | continuous_map.norm_eq_supr_norm | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_restrict_mono_set {X : Type*} [topological_space X]
(f : C(X, E)) {K L : topological_space.compacts X} (hKL : K ≤ L) :
‖f.restrict K‖ ≤ ‖f.restrict L‖ | (norm_le _ (norm_nonneg _)).mpr (λ x, norm_coe_le_norm (f.restrict L) $ set.inclusion hKL x) | lemma | continuous_map.norm_restrict_mono_set | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"set.inclusion",
"topological_space",
"topological_space.compacts"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_bounded_of_compact :
C(α, E) ≃ₗᵢ[𝕜] (α →ᵇ E) | { map_smul' := λ c f, by { ext, simp, },
norm_map' := λ f, rfl,
.. add_equiv_bounded_of_compact α E } | def | continuous_map.linear_isometry_bounded_of_compact | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_clm (x : α) : C(α, E) →L[𝕜] E | (eval_clm 𝕜 x).comp
((linear_isometry_bounded_of_compact α E 𝕜).to_linear_isometry).to_continuous_linear_map | def | continuous_map.eval_clm | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_bounded_of_compact_symm_apply (f : α →ᵇ E) :
(linear_isometry_bounded_of_compact α E 𝕜).symm f = f.to_continuous_map | rfl | lemma | continuous_map.linear_isometry_bounded_of_compact_symm_apply | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_bounded_of_compact_apply_apply (f : C(α, E)) (a : α) :
(linear_isometry_bounded_of_compact α E 𝕜 f) a = f a | rfl | lemma | continuous_map.linear_isometry_bounded_of_compact_apply_apply | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_bounded_of_compact_to_isometry_equiv :
(linear_isometry_bounded_of_compact α E 𝕜).to_isometry_equiv =
(isometry_equiv_bounded_of_compact α E) | rfl | lemma | continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_bounded_of_compact_to_add_equiv :
(linear_isometry_bounded_of_compact α E 𝕜).to_linear_equiv.to_add_equiv =
(add_equiv_bounded_of_compact α E) | rfl | lemma | continuous_map.linear_isometry_bounded_of_compact_to_add_equiv | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry_bounded_of_compact_of_compact_to_equiv :
(linear_isometry_bounded_of_compact α E 𝕜).to_linear_equiv.to_equiv =
(equiv_bounded_of_compact α E) | rfl | lemma | continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuity
(f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε | metric.uniform_continuous_iff.mp
(compact_space.uniform_continuous_of_continuous f.continuous) ε h | lemma | continuous_map.uniform_continuity | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"compact_space.uniform_continuous_of_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ | classical.some (uniform_continuity f ε h) | def | continuous_map.modulus | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h | (classical.some_spec (uniform_continuity f ε h)).fst | lemma | continuous_map.modulus_pos | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_lt_of_dist_lt_modulus
(f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α} (w : dist a b < f.modulus ε h) :
dist (f a) (f b) < ε | (classical.some_spec (uniform_continuity f ε h)).snd w | lemma | continuous_map.dist_lt_of_dist_lt_modulus | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.comp_left_continuous_compact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) | (linear_isometry_bounded_of_compact X γ 𝕜).symm.to_linear_isometry.to_continuous_linear_map.comp $
(g.comp_left_continuous_bounded X).comp $
(linear_isometry_bounded_of_compact X β 𝕜).to_linear_isometry.to_continuous_linear_map | def | continuous_linear_map.comp_left_continuous_compact | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `continuous_linear_map.comp_left_continuous_bounded`,
upgraded version of `continuous_linear_map.comp_left_continuous`,
similar to `linear_map.comp_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.to_linear_comp_left_continuous_compact (g : β →L[𝕜] γ) :
(g.comp_left_continuous_compact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.comp_left_continuous 𝕜 X | by { ext f, refl } | lemma | continuous_linear_map.to_linear_comp_left_continuous_compact | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.comp_left_continuous_compact_apply (g : β →L[𝕜] γ)
(f : C(X, β)) (x : X) :
g.comp_left_continuous_compact X f x = g (f x) | rfl | lemma | continuous_linear_map.comp_left_continuous_compact_apply | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_continuous_map {X Y : Type*} (T : Type*) [topological_space X] [compact_space X]
[topological_space Y] [compact_space Y] [metric_space T]
(f : C(X, Y)) : C(C(Y, T), C(X, T)) | { to_fun := λ g, g.comp f,
continuous_to_fun :=
begin
refine metric.continuous_iff.mpr _,
intros g ε ε_pos,
refine ⟨ε, ε_pos, λ g' h, _⟩,
rw continuous_map.dist_lt_iff ε_pos at h ⊢,
{ exact λ x, h (f x), },
end } | def | continuous_map.comp_right_continuous_map | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"compact_space",
"continuous_map.dist_lt_iff",
"metric_space",
"topological_space"
] | Precomposition by a continuous map is itself a continuous map between spaces of continuous maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_right_continuous_map_apply {X Y : Type*} (T : Type*) [topological_space X]
[compact_space X] [topological_space Y] [compact_space Y] [metric_space T]
(f : C(X, Y)) (g : C(Y, T)) :
(comp_right_continuous_map T f) g = g.comp f | rfl | lemma | continuous_map.comp_right_continuous_map_apply | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"compact_space",
"metric_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right_homeomorph {X Y : Type*} (T : Type*) [topological_space X] [compact_space X]
[topological_space Y] [compact_space Y] [metric_space T]
(f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) | { to_fun := comp_right_continuous_map T f.to_continuous_map,
inv_fun := comp_right_continuous_map T f.symm.to_continuous_map,
left_inv := λ g, ext $ λ _, congr_arg g (f.apply_symm_apply _),
right_inv := λ g, ext $ λ _, congr_arg g (f.symm_apply_apply _) } | def | continuous_map.comp_right_homeomorph | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"compact_space",
"inv_fun",
"metric_space",
"topological_space"
] | Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_right_alg_hom_continuous {X Y : Type*} (R A : Type*)
[topological_space X] [compact_space X] [topological_space Y] [compact_space Y] [comm_semiring R]
[semiring A] [metric_space A] [topological_semiring A] [algebra R A] (f : C(X, Y)) :
continuous (comp_right_alg_hom R A f) | map_continuous (comp_right_continuous_map A f) | lemma | continuous_map.comp_right_alg_hom_continuous | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"algebra",
"comm_semiring",
"compact_space",
"continuous",
"metric_space",
"semiring",
"topological_semiring",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : compacts X, summable (λ i, ‖(F i).restrict K‖)) :
summable F | begin
refine (continuous_map.exists_tendsto_compact_open_iff_forall _).2 (λ K hK, _),
lift K to compacts X using hK,
have A : ∀ s : finset ι, restrict ↑K (∑ i in s, F i) = ∑ i in s, restrict K (F i),
{ intro s, ext1 x, simp },
simpa only [has_sum, A] using summable_of_summable_norm (hF K)
end | lemma | continuous_map.summable_of_locally_summable_norm | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"continuous_map.exists_tendsto_compact_open_iff_forall",
"finset",
"has_sum",
"lift",
"summable",
"summable_of_summable_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.bounded_continuous_function.mk_of_compact_star [compact_space α] (f : C(α, β)) :
mk_of_compact (star f) = star (mk_of_compact f) | rfl | lemma | bounded_continuous_function.mk_of_compact_star | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_of_set (s : set X) : ideal C(X, R) | { carrier := {f : C(X, R) | ∀ x ∈ sᶜ, f x = 0},
add_mem' := λ f g hf hg x hx, by simp only [hf x hx, hg x hx, coe_add, pi.add_apply, add_zero],
zero_mem' := λ _ _, rfl,
smul_mem' := λ c f hf x hx, mul_zero (c x) ▸ congr_arg (λ y, c x * y) (hf x hx), } | def | continuous_map.ideal_of_set | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"ideal",
"mul_zero"
] | Given a topological ring `R` and `s : set X`, construct the ideal in `C(X, R)` of functions
which vanish on the complement of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_of_set_closed [locally_compact_space X] [t2_space R] (s : set X) :
is_closed (ideal_of_set R s : set C(X, R) ) | begin
simp only [ideal_of_set, submodule.coe_set_mk, set.set_of_forall],
exact is_closed_Inter (λ x, is_closed_Inter $
λ hx, is_closed_eq (continuous_eval_const' x) continuous_const),
end | lemma | continuous_map.ideal_of_set_closed | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"continuous_const",
"is_closed",
"is_closed_Inter",
"is_closed_eq",
"locally_compact_space",
"set.set_of_forall",
"submodule.coe_set_mk",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ideal_of_set {s : set X} {f : C(X, R)} :
f ∈ ideal_of_set R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0 | iff.rfl | lemma | continuous_map.mem_ideal_of_set | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_mem_ideal_of_set {s : set X} {f : C(X, R)} :
f ∉ ideal_of_set R s ↔ ∃ x ∈ sᶜ, f x ≠ 0 | by { simp_rw [mem_ideal_of_set, exists_prop], push_neg } | lemma | continuous_map.not_mem_ideal_of_set | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_ideal (I : ideal C(X, R)) : set X | {x : X | ∀ f ∈ I, (f : C(X, R)) x = 0}ᶜ | def | continuous_map.set_of_ideal | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"ideal"
] | Given an ideal `I` of `C(X, R)`, construct the set of points for which every function in the
ideal vanishes on the complement. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_mem_set_of_ideal {I : ideal C(X, R)} {x : X} :
x ∉ set_of_ideal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0 | by rw [←set.mem_compl_iff, set_of_ideal, compl_compl, set.mem_set_of] | lemma | continuous_map.not_mem_set_of_ideal | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"compl_compl",
"ideal",
"set.mem_set_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_set_of_ideal {I : ideal C(X, R)} {x : X} :
x ∈ set_of_ideal I ↔ ∃ f ∈ I, (f : C(X, R)) x ≠ 0 | by { simp_rw [set_of_ideal, set.mem_compl_iff, set.mem_set_of, exists_prop], push_neg } | lemma | continuous_map.mem_set_of_ideal | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"exists_prop",
"ideal",
"set.mem_compl_iff",
"set.mem_set_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_ideal_open [t2_space R] (I : ideal C(X, R)) : is_open (set_of_ideal I) | begin
simp only [set_of_ideal, set.set_of_forall, is_open_compl_iff],
exact is_closed_Inter (λ f, is_closed_Inter $
λ hf, is_closed_eq (map_continuous f) continuous_const)
end | lemma | continuous_map.set_of_ideal_open | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"continuous_const",
"ideal",
"is_closed_Inter",
"is_closed_eq",
"is_open",
"is_open_compl_iff",
"set.set_of_forall",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opens_of_ideal [t2_space R] (I : ideal C(X, R)) : opens X | ⟨set_of_ideal I, set_of_ideal_open I⟩ | def | continuous_map.opens_of_ideal | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"ideal",
"t2_space"
] | The open set `set_of_ideal I` realized as a term of `opens X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_of_top_eq_univ [nontrivial R] : (set_of_ideal (⊤ : ideal C(X, R))) = set.univ | set.univ_subset_iff.mp $ λ x hx, mem_set_of_ideal.mpr ⟨1, submodule.mem_top, one_ne_zero⟩ | lemma | continuous_map.set_of_top_eq_univ | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"ideal",
"nontrivial",
"submodule.mem_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_of_empty_eq_bot : (ideal_of_set R (∅ : set X)) = ⊥ | ideal.ext (λ f, by simpa only [mem_ideal_of_set, set.compl_empty, set.mem_univ, forall_true_left,
ideal.mem_bot, fun_like.ext_iff] using iff.rfl) | lemma | continuous_map.ideal_of_empty_eq_bot | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"forall_true_left",
"fun_like.ext_iff",
"ideal.ext",
"ideal.mem_bot",
"set.compl_empty",
"set.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ideal_of_set_compl_singleton (x : X) (f : C(X, R)) :
f ∈ ideal_of_set R ({x}ᶜ : set X) ↔ f x = 0 | by simp only [mem_ideal_of_set, compl_compl, set.mem_singleton_iff, forall_eq] | lemma | continuous_map.mem_ideal_of_set_compl_singleton | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"compl_compl",
"forall_eq",
"set.mem_singleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_gc : galois_connection (set_of_ideal : ideal C(X, R) → set X) (ideal_of_set R) | begin
refine λ I s, ⟨λ h f hf, _, λ h x hx, _⟩,
{ by_contra h',
rcases not_mem_ideal_of_set.mp h' with ⟨x, hx, hfx⟩,
exact hfx (not_mem_set_of_ideal.mp (mt (@h x) hx) hf) },
{ obtain ⟨f, hf, hfx⟩ := mem_set_of_ideal.mp hx,
by_contra hx',
exact not_mem_ideal_of_set.mpr ⟨x, hx', hfx⟩ (h hf) },
end | lemma | continuous_map.ideal_gc | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"by_contra",
"galois_connection",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mul_le_one_eq_on_ge (f : C(X, ℝ≥0)) {c : ℝ≥0} (hc : 0 < c) :
∃ g : C(X, ℝ≥0), (∀ x : X, (g * f) x ≤ 1) ∧ {x : X | c ≤ f x}.eq_on (g * f) 1 | ⟨{ to_fun := (f ⊔ (const X c))⁻¹,
continuous_to_fun := ((map_continuous f).sup $ map_continuous _).inv₀
(λ _, (hc.trans_le le_sup_right).ne')},
λ x, (inv_mul_le_iff (hc.trans_le le_sup_right)).mpr ((mul_one (f x ⊔ c)).symm ▸ le_sup_left),
λ x hx, by simpa only [coe_const, coe_mk, pi.mul_apply, pi.inv_apply, p... | lemma | continuous_map.exists_mul_le_one_eq_on_ge | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"function.const_apply",
"inv_mul_cancel",
"inv_mul_le_iff",
"le_sup_left",
"le_sup_right",
"mul_one",
"pi.inv_apply",
"pi.mul_apply",
"pi.one_apply",
"pi.sup_apply"
] | An auxiliary lemma used in the proof of `ideal_of_set_of_ideal_eq_closure` which may be useful
on its own. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_of_set_of_ideal_eq_closure (I : ideal C(X, 𝕜)) :
ideal_of_set 𝕜 (set_of_ideal I) = I.closure | begin
/- Since `ideal_of_set 𝕜 (set_of_ideal I)` is closed and contains `I`, it contains `I.closure`.
For the reverse inclusion, given `f ∈ ideal_of_set 𝕜 (set_of_ideal I)` and `(ε : ℝ≥0) > 0` it
suffices to show that `f` is within `ε` of `I`.-/
refine le_antisymm (λ f hf, metric.mem_closure_iff.mpr (λ ε hε, ... | lemma | continuous_map.ideal_of_set_of_ideal_eq_closure | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"algebra_map_clm",
"algebra_map_clm_coe",
"compl_compl",
"continuous_at.eventually_ne",
"continuous_const",
"continuous_map.coe_coe",
"continuous_map.coe_comp",
"continuous_on",
"disjoint",
"dist_nndist",
"half_pos",
"ideal",
"is_R_or_C.c... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_of_set_of_ideal_is_closed {I : ideal C(X, 𝕜)}
(hI : is_closed (I : set C(X, 𝕜))) : ideal_of_set 𝕜 (set_of_ideal I) = I | (ideal_of_set_of_ideal_eq_closure I).trans (ideal.ext $ set.ext_iff.mp hI.closure_eq) | lemma | continuous_map.ideal_of_set_of_ideal_is_closed | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"ideal",
"ideal.ext",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_ideal_of_set_eq_interior (s : set X) :
set_of_ideal (ideal_of_set 𝕜 s) = interior s | begin
refine set.subset.antisymm ((set_of_ideal_open (ideal_of_set 𝕜 s)).subset_interior_iff.mpr
(λ x hx, let ⟨f, hf, hfx⟩ := mem_set_of_ideal.mp hx
in set.not_mem_compl_iff.mp (mt (@hf x) hfx))) (λ x hx, _),
/- If `x ∉ closure sᶜ`, we must produce `f : C(X, 𝕜)` which is zero on `sᶜ` and `f x ≠ 0`. -/
r... | lemma | continuous_map.set_of_ideal_of_set_eq_interior | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"exists_continuous_zero_one_of_closed",
"interior",
"is_R_or_C.of_real_one",
"is_closed_closure",
"is_closed_singleton",
"normal_of_compact_t2",
"normal_space",
"pi.one_apply",
"set.eq_on",
"set.mem_singleton",
"set.subset.antisymm",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_ideal_of_set_of_is_open {s : set X} (hs : is_open s) :
set_of_ideal (ideal_of_set 𝕜 s) = s | (set_of_ideal_of_set_eq_interior 𝕜 s).trans hs.interior_eq | lemma | continuous_map.set_of_ideal_of_set_of_is_open | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_opens_gi :
galois_insertion (opens_of_ideal : ideal C(X, 𝕜) → opens X) (λ s, ideal_of_set 𝕜 s) | { choice := λ I hI, opens_of_ideal I.closure,
gc := λ I s, ideal_gc X 𝕜 I s,
le_l_u := λ s, (set_of_ideal_of_set_of_is_open 𝕜 s.is_open).ge,
choice_eq := λ I hI, congr_arg _ $ ideal.ext (set.ext_iff.mp (is_closed_of_closure_subset $
(ideal_of_set_of_ideal_eq_closure I ▸ hI : I.closure ≤ I)).closure_eq) } | def | continuous_map.ideal_opens_gi | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"galois_insertion",
"ideal",
"ideal.ext",
"is_closed_of_closure_subset"
] | The Galois insertion `continuous_map.opens_of_ideal : ideal C(X, 𝕜) → opens X` and
`λ s, continuous_map.ideal_of_set ↑s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_of_set_is_maximal_iff (s : opens X) :
(ideal_of_set 𝕜 (s : set X)).is_maximal ↔ is_coatom s | begin
rw ideal.is_maximal_def,
refine (ideal_opens_gi X 𝕜).is_coatom_iff (λ I hI, _) s,
rw ←ideal.is_maximal_def at hI,
resetI,
exact ideal_of_set_of_ideal_is_closed infer_instance,
end | lemma | continuous_map.ideal_of_set_is_maximal_iff | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"ideal.is_maximal_def",
"is_coatom",
"is_coatom_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_of_compl_singleton_is_maximal (x : X) : (ideal_of_set 𝕜 ({x}ᶜ : set X)).is_maximal | (ideal_of_set_is_maximal_iff 𝕜 (closeds.singleton x).compl).mpr $ opens.is_coatom_iff.mpr ⟨x, rfl⟩ | lemma | continuous_map.ideal_of_compl_singleton_is_maximal | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_ideal_eq_compl_singleton (I : ideal C(X, 𝕜)) [hI : I.is_maximal] :
∃ x : X, set_of_ideal I = {x}ᶜ | begin
have h : (ideal_of_set 𝕜 (set_of_ideal I)).is_maximal, from
(ideal_of_set_of_ideal_is_closed (infer_instance : is_closed (I : set C(X, 𝕜)))).symm ▸ hI,
obtain ⟨x, hx⟩ := opens.is_coatom_iff.1 ((ideal_of_set_is_maximal_iff 𝕜 (opens_of_ideal I)).1 h),
exact ⟨x, congr_arg coe hx⟩,
end | lemma | continuous_map.set_of_ideal_eq_compl_singleton | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"ideal",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_is_maximal_iff (I : ideal C(X, 𝕜)) [hI : is_closed (I : set C(X, 𝕜))] :
I.is_maximal ↔ ∃ x : X, ideal_of_set 𝕜 {x}ᶜ = I | begin
refine ⟨_, λ h, let ⟨x, hx⟩ := h in hx ▸ ideal_of_compl_singleton_is_maximal 𝕜 x⟩,
introI hI',
obtain ⟨x, hx⟩ := set_of_ideal_eq_compl_singleton I,
exact ⟨x, by simpa only [ideal_of_set_of_ideal_eq_closure, ideal.closure_eq_of_is_closed]
using congr_arg (ideal_of_set 𝕜) hx.symm⟩,
end | lemma | continuous_map.ideal_is_maximal_iff | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"ideal",
"ideal.closure_eq_of_is_closed",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_map_eval :
C(X, character_space 𝕜 C(X, 𝕜)) | { to_fun := λ x, ⟨{ to_fun := λ f, f x, map_add' := λ f g, rfl, map_smul' := λ z f, rfl,
cont := continuous_eval_const' x },
by { rw character_space.eq_set_map_one_map_mul, exact ⟨rfl, λ f g, rfl⟩ }⟩,
continuous_to_fun := continuous.subtype_mk (continuous_of_continuous_eval map_c... | def | weak_dual.character_space.continuous_map_eval | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"cont",
"continuous.subtype_mk"
] | The natural continuous map from a locally compact topological space `X` to the
`character_space 𝕜 C(X, 𝕜)` which sends `x : X` to point evaluation at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map_eval_apply_apply (x : X) (f : C(X, 𝕜)) :
continuous_map_eval X 𝕜 x f = f x | rfl | lemma | weak_dual.character_space.continuous_map_eval_apply_apply | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_map_eval_bijective : bijective (continuous_map_eval X 𝕜) | begin
refine ⟨λ x y hxy, _, λ φ, _⟩,
{ contrapose! hxy,
haveI := @normal_of_compact_t2 X _ _ _,
rcases exists_continuous_zero_one_of_closed (is_closed_singleton : _root_.is_closed {x})
(is_closed_singleton : _root_.is_closed {y}) (set.disjoint_singleton.mpr hxy)
with ⟨f, fx, fy, -⟩,
rw [←ne.... | lemma | weak_dual.character_space.continuous_map_eval_bijective | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"continuous_map.comp_apply",
"exists_continuous_zero_one_of_closed",
"fun_like.ne_iff",
"ideal.ext",
"is_R_or_C.of_real_inj",
"is_closed_singleton",
"normal_of_compact_t2",
"ring_hom.ker",
"ring_hom.mem_ker",
"set.mem_singleton",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeo_eval : X ≃ₜ character_space 𝕜 C(X, 𝕜) | @continuous.homeo_of_equiv_compact_to_t2 _ _ _ _ _ _
{ to_fun := (continuous_map_eval X 𝕜),
.. equiv.of_bijective _ (continuous_map_eval_bijective X 𝕜) }
(map_continuous (continuous_map_eval X 𝕜)) | def | weak_dual.character_space.homeo_eval | topology.continuous_function | src/topology/continuous_function/ideals.lean | [
"topology.algebra.algebra",
"topology.continuous_function.compact",
"topology.urysohns_lemma",
"data.is_R_or_C.basic",
"analysis.normed_space.units",
"topology.algebra.module.character_space"
] | [
"continuous.homeo_of_equiv_compact_to_t2",
"equiv.of_bijective"
] | This is the natural homeomorphism between a compact Hausdorff space `X` and the
`character_space 𝕜 C(X, 𝕜)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_map_monoid_hom [monoid Y] [has_continuous_mul Y] :
locally_constant X Y →* C(X, Y) | { to_fun := coe,
map_one' := by { ext, simp, },
map_mul' := λ x y, by { ext, simp, }, } | def | locally_constant.to_continuous_map_monoid_hom | topology.continuous_function | src/topology/continuous_function/locally_constant.lean | [
"topology.locally_constant.algebra",
"topology.continuous_function.basic",
"topology.continuous_function.algebra"
] | [
"has_continuous_mul",
"locally_constant",
"monoid"
] | The inclusion of locally-constant functions into continuous functions as a multiplicative
monoid hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_map_linear_map (R : Type*) [semiring R]
[add_comm_monoid Y] [module R Y] [has_continuous_add Y] [has_continuous_const_smul R Y] :
locally_constant X Y →ₗ[R] C(X, Y) | { to_fun := coe,
map_add' := λ x y, by { ext, simp, },
map_smul' := λ x y, by { ext, simp, }, } | def | locally_constant.to_continuous_map_linear_map | topology.continuous_function | src/topology/continuous_function/locally_constant.lean | [
"topology.locally_constant.algebra",
"topology.continuous_function.basic",
"topology.continuous_function.algebra"
] | [
"add_comm_monoid",
"has_continuous_add",
"has_continuous_const_smul",
"locally_constant",
"module",
"semiring"
] | The inclusion of locally-constant functions into continuous functions as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_map_alg_hom (R : Type*) [comm_semiring R]
[semiring Y] [algebra R Y] [topological_semiring Y] :
locally_constant X Y →ₐ[R] C(X, Y) | { to_fun := coe,
map_one' := by { ext, simp, },
map_mul' := λ x y, by { ext, simp, },
map_zero' := by { ext, simp, },
map_add' := λ x y, by { ext, simp, },
commutes' := λ r, by { ext x, simp [algebra.smul_def], }, } | def | locally_constant.to_continuous_map_alg_hom | topology.continuous_function | src/topology/continuous_function/locally_constant.lean | [
"topology.locally_constant.algebra",
"topology.continuous_function.basic",
"topology.continuous_function.algebra"
] | [
"algebra",
"algebra.smul_def",
"comm_semiring",
"locally_constant",
"semiring",
"topological_semiring"
] | The inclusion of locally-constant functions into continuous functions as an algebra map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs (f : C(α, β)) : C(α, β) | { to_fun := λ x, |f x|, } | def | continuous_map.abs | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [] | The pointwise absolute value of a continuous function as a continuous function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_apply (f : C(α, β)) (x : α) : |f| x = |f x| | rfl | lemma | continuous_map.abs_apply | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
partial_order [partial_order β] :
partial_order C(α, β) | partial_order.lift (λ f, f.to_fun) (by tidy) | instance | continuous_map.partial_order | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"partial_order.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_def [partial_order β] {f g : C(α, β)} : f ≤ g ↔ ∀ a, f a ≤ g a | pi.le_def | lemma | continuous_map.le_def | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"pi.le_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_def [partial_order β] {f g : C(α, β)} :
f < g ↔ (∀ a, f a ≤ g a) ∧ (∃ a, f a < g a) | pi.lt_def | lemma | continuous_map.lt_def | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"pi.lt_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sup [linear_order β] [order_closed_topology β] : has_sup C(α, β) | { sup := λ f g, { to_fun := λ a, max (f a) (g a), } } | instance | continuous_map.has_sup | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"has_sup",
"order_closed_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_coe [linear_order β] [order_closed_topology β] (f g : C(α, β)) :
((f ⊔ g : C(α, β)) : α → β) = (f ⊔ g : α → β) | rfl | lemma | continuous_map.sup_coe | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"order_closed_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_apply [linear_order β] [order_closed_topology β] (f g : C(α, β)) (a : α) :
(f ⊔ g) a = max (f a) (g a) | rfl | lemma | continuous_map.sup_apply | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"order_closed_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_inf [linear_order β] [order_closed_topology β] : has_inf C(α, β) | { inf := λ f g, { to_fun := λ a, min (f a) (g a), } } | instance | continuous_map.has_inf | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"has_inf",
"order_closed_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_coe [linear_order β] [order_closed_topology β] (f g : C(α, β)) :
((f ⊓ g : C(α, β)) : α → β) = (f ⊓ g : α → β) | rfl | lemma | continuous_map.inf_coe | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"order_closed_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_apply [linear_order β] [order_closed_topology β] (f g : C(α, β)) (a : α) :
(f ⊓ g) a = min (f a) (g a) | rfl | lemma | continuous_map.inf_apply | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"order_closed_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup'_apply {ι : Type*} {s : finset ι} (H : s.nonempty) (f : ι → C(β, γ)) (b : β) :
s.sup' H f b = s.sup' H (λ a, f a b) | finset.comp_sup'_eq_sup'_comp H (λ f : C(β, γ), f b) (λ i j, rfl) | lemma | continuous_map.sup'_apply | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"finset",
"finset.comp_sup'_eq_sup'_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup'_coe {ι : Type*} {s : finset ι} (H : s.nonempty) (f : ι → C(β, γ)) :
((s.sup' H f : C(β, γ)) : ι → β) = s.sup' H (λ a, (f a : β → γ)) | by { ext, simp [sup'_apply], } | lemma | continuous_map.sup'_coe | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf'_apply {ι : Type*} {s : finset ι} (H : s.nonempty) (f : ι → C(β, γ)) (b : β) :
s.inf' H f b = s.inf' H (λ a, f a b) | @sup'_apply _ γᵒᵈ _ _ _ _ _ _ H f b | lemma | continuous_map.inf'_apply | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf'_coe {ι : Type*} {s : finset ι} (H : s.nonempty) (f : ι → C(β, γ)) :
((s.inf' H f : C(β, γ)) : ι → β) = s.inf' H (λ a, (f a : β → γ)) | @sup'_coe _ γᵒᵈ _ _ _ _ _ _ H f | lemma | continuous_map.inf'_coe | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Icc_extend (f : C(set.Icc a b, β)) : C(α, β) | ⟨set.Icc_extend h f⟩ | def | continuous_map.Icc_extend | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"set.Icc"
] | Extend a continuous function `f : C(set.Icc a b, β)` to a function `f : C(α, β)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_Icc_extend (f : C(set.Icc a b, β)) :
((Icc_extend h f : C(α, β)) : α → β) = set.Icc_extend h f | rfl | lemma | continuous_map.coe_Icc_extend | topology.continuous_function | src/topology/continuous_function/ordered.lean | [
"topology.algebra.order.proj_Icc",
"topology.algebra.order.group",
"topology.continuous_function.basic"
] | [
"set.Icc",
"set.Icc_extend"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map (p : R[X]) : C(R, R) | ⟨λ x : R, p.eval x, by continuity⟩ | def | polynomial.to_continuous_map | topology.continuous_function | src/topology/continuous_function/polynomial.lean | [
"topology.algebra.polynomial",
"topology.continuous_function.algebra",
"topology.unit_interval"
] | [] | Every polynomial with coefficients in a topological semiring gives a (bundled) continuous function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_map_on (p : R[X]) (X : set R) : C(X, R) | ⟨λ x : X, p.to_continuous_map x, by continuity⟩ | def | polynomial.to_continuous_map_on | topology.continuous_function | src/topology/continuous_function/polynomial.lean | [
"topology.algebra.polynomial",
"topology.continuous_function.algebra",
"topology.unit_interval"
] | [] | A polynomial as a continuous function,
with domain restricted to some subset of the semiring of coefficients.
(This is particularly useful when restricting to compact sets, e.g. `[0,1]`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aeval_continuous_map_apply (g : R[X]) (f : C(α, R)) (x : α) :
((polynomial.aeval f) g) x = g.eval (f x) | begin
apply polynomial.induction_on' g,
{ intros p q hp hq, simp [hp, hq], },
{ intros n a, simp [pi.pow_apply], },
end | lemma | polynomial.aeval_continuous_map_apply | topology.continuous_function | src/topology/continuous_function/polynomial.lean | [
"topology.algebra.polynomial",
"topology.continuous_function.algebra",
"topology.unit_interval"
] | [
"pi.pow_apply",
"polynomial.aeval",
"polynomial.induction_on'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map_alg_hom : R[X] →ₐ[R] C(R, R) | { to_fun := λ p, p.to_continuous_map,
map_zero' := by { ext, simp, },
map_add' := by { intros, ext, simp, },
map_one' := by { ext, simp, },
map_mul' := by { intros, ext, simp, },
commutes' := by { intros, ext, simp [algebra.algebra_map_eq_smul_one], }, } | def | polynomial.to_continuous_map_alg_hom | topology.continuous_function | src/topology/continuous_function/polynomial.lean | [
"topology.algebra.polynomial",
"topology.continuous_function.algebra",
"topology.unit_interval"
] | [
"algebra.algebra_map_eq_smul_one"
] | The algebra map from `R[X]` to continuous functions `C(R, R)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_map_on_alg_hom (X : set R) : R[X] →ₐ[R] C(X, R) | { to_fun := λ p, p.to_continuous_map_on X,
map_zero' := by { ext, simp, },
map_add' := by { intros, ext, simp, },
map_one' := by { ext, simp, },
map_mul' := by { intros, ext, simp, },
commutes' := by { intros, ext, simp [algebra.algebra_map_eq_smul_one], }, } | def | polynomial.to_continuous_map_on_alg_hom | topology.continuous_function | src/topology/continuous_function/polynomial.lean | [
"topology.algebra.polynomial",
"topology.continuous_function.algebra",
"topology.unit_interval"
] | [
"algebra.algebra_map_eq_smul_one"
] | The algebra map from `R[X]` to continuous functions `C(X, R)`, for any subset `X` of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial_functions (X : set R) : subalgebra R C(X, R) | (⊤ : subalgebra R R[X]).map (polynomial.to_continuous_map_on_alg_hom X) | def | polynomial_functions | topology.continuous_function | src/topology/continuous_function/polynomial.lean | [
"topology.algebra.polynomial",
"topology.continuous_function.algebra",
"topology.unit_interval"
] | [
"polynomial.to_continuous_map_on_alg_hom",
"subalgebra"
] | The subalgebra of polynomial functions in `C(X, R)`, for `X` a subset of some topological semiring
`R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial_functions_coe (X : set R) :
(polynomial_functions X : set C(X, R)) = set.range (polynomial.to_continuous_map_on_alg_hom X) | by { ext, simp [polynomial_functions], } | lemma | polynomial_functions_coe | topology.continuous_function | src/topology/continuous_function/polynomial.lean | [
"topology.algebra.polynomial",
"topology.continuous_function.algebra",
"topology.unit_interval"
] | [
"polynomial.to_continuous_map_on_alg_hom",
"polynomial_functions",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial_functions_separates_points (X : set R) :
(polynomial_functions X).separates_points | λ x y h,
begin
-- We use `polynomial.X`, then clean up.
refine ⟨_, ⟨⟨_, ⟨⟨polynomial.X, ⟨algebra.mem_top, rfl⟩⟩, rfl⟩⟩, _⟩⟩,
dsimp, simp only [polynomial.eval_X],
exact (λ h', h (subtype.ext h')),
end | lemma | polynomial_functions_separates_points | topology.continuous_function | src/topology/continuous_function/polynomial.lean | [
"topology.algebra.polynomial",
"topology.continuous_function.algebra",
"topology.unit_interval"
] | [
"polynomial.eval_X",
"polynomial_functions",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial_functions.comap_comp_right_alg_hom_Icc_homeo_I (a b : ℝ) (h : a < b) :
(polynomial_functions I).comap
(comp_right_alg_hom ℝ ℝ (Icc_homeo_I a b h).symm.to_continuous_map) =
polynomial_functions (set.Icc a b) | begin
ext f,
fsplit,
{ rintro ⟨p, ⟨-,w⟩⟩,
rw fun_like.ext_iff at w,
dsimp at w,
let q := p.comp ((b - a)⁻¹ • polynomial.X + polynomial.C (-a * (b-a)⁻¹)),
refine ⟨q, ⟨_, _⟩⟩,
{ simp, },
{ ext x,
simp only [neg_mul,
ring_hom.map_neg, ring_hom.map_mul, alg_hom.coe_to_ring_hom,
... | lemma | polynomial_functions.comap_comp_right_alg_hom_Icc_homeo_I | topology.continuous_function | src/topology/continuous_function/polynomial.lean | [
"topology.algebra.polynomial",
"topology.continuous_function.algebra",
"topology.unit_interval"
] | [
"Icc_homeo_I",
"Icc_homeo_I_symm_apply_coe",
"alg_hom.coe_to_ring_hom",
"div_le_one",
"fun_like.ext_iff",
"mul_comm",
"neg_mul",
"polynomial.C",
"polynomial.X",
"polynomial.coe_aeval_eq_eval",
"polynomial.eval_C",
"polynomial.eval_X",
"polynomial.eval_add",
"polynomial.eval_comp",
"polyn... | The preimage of polynomials on `[0,1]` under the pullback map by `x ↦ (b-a) * x + a`
is the polynomials on `[a,b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
attach_bound (f : C(X, ℝ)) : C(X, set.Icc (-‖f‖) (‖f‖)) | { to_fun := λ x, ⟨f x, ⟨neg_norm_le_apply f x, apply_le_norm f x⟩⟩ } | def | continuous_map.attach_bound | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"set.Icc"
] | Turn a function `f : C(X, ℝ)` into a continuous map into `set.Icc (-‖f‖) (‖f‖)`,
thereby explicitly attaching bounds. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
attach_bound_apply_coe (f : C(X, ℝ)) (x : X) : ((attach_bound f) x : ℝ) = f x | rfl | lemma | continuous_map.attach_bound_apply_coe | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial_comp_attach_bound (A : subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
(g.to_continuous_map_on (set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attach_bound =
polynomial.aeval f g | begin
ext,
simp only [continuous_map.coe_comp, function.comp_app,
continuous_map.attach_bound_apply_coe,
polynomial.to_continuous_map_on_apply,
polynomial.aeval_subalgebra_coe,
polynomial.aeval_continuous_map_apply,
polynomial.to_continuous_map_apply],
end | lemma | continuous_map.polynomial_comp_attach_bound | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"continuous_map.attach_bound_apply_coe",
"continuous_map.coe_comp",
"polynomial.aeval",
"polynomial.aeval_continuous_map_apply",
"polynomial.aeval_subalgebra_coe",
"set.Icc",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial_comp_attach_bound_mem (A : subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
(g.to_continuous_map_on (set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attach_bound ∈ A | begin
rw polynomial_comp_attach_bound,
apply set_like.coe_mem,
end | lemma | continuous_map.polynomial_comp_attach_bound_mem | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"set.Icc",
"set_like.coe_mem",
"subalgebra"
] | Given a continuous function `f` in a subalgebra of `C(X, ℝ)`, postcomposing by a polynomial
gives another function in `A`.
This lemma proves something slightly more subtle than this:
we take `f`, and think of it as a function into the restricted target `set.Icc (-‖f‖) ‖f‖)`,
and then postcompose with a polynomial func... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_attach_bound_mem_closure
(A : subalgebra ℝ C(X, ℝ)) (f : A) (p : C(set.Icc (-‖f‖) (‖f‖), ℝ)) :
p.comp (attach_bound f) ∈ A.topological_closure | begin
-- `p` itself is in the closure of polynomials, by the Weierstrass theorem,
have mem_closure : p ∈ (polynomial_functions (set.Icc (-‖f‖) (‖f‖))).topological_closure :=
continuous_map_mem_polynomial_functions_closure _ _ p,
-- and so there are polynomials arbitrarily close.
have frequently_mem_polynomi... | theorem | continuous_map.comp_attach_bound_mem_closure | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"alg_hom.coe_to_ring_hom",
"continuous_at",
"continuous_map_mem_polynomial_functions_closure",
"polynomial_functions",
"set.Icc",
"set_like.mem_coe",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_mem_subalgebra_closure (A : subalgebra ℝ C(X, ℝ)) (f : A) :
(f : C(X, ℝ)).abs ∈ A.topological_closure | begin
let M := ‖f‖,
let f' := attach_bound (f : C(X, ℝ)),
let abs : C(set.Icc (-‖f‖) (‖f‖), ℝ) :=
{ to_fun := λ x : set.Icc (-‖f‖) (‖f‖), |(x : ℝ)| },
change (abs.comp f') ∈ A.topological_closure,
apply comp_attach_bound_mem_closure,
end | theorem | continuous_map.abs_mem_subalgebra_closure | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"set.Icc",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_mem_subalgebra_closure (A : subalgebra ℝ C(X, ℝ)) (f g : A) :
(f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A.topological_closure | begin
rw inf_eq,
refine A.topological_closure.smul_mem
(A.topological_closure.sub_mem
(A.topological_closure.add_mem (A.le_topological_closure f.property)
(A.le_topological_closure g.property)) _) _,
exact_mod_cast abs_mem_subalgebra_closure A _,
end | theorem | continuous_map.inf_mem_subalgebra_closure | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_mem_closed_subalgebra (A : subalgebra ℝ C(X, ℝ)) (h : is_closed (A : set C(X, ℝ)))
(f g : A) : (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A | begin
convert inf_mem_subalgebra_closure A f g,
apply set_like.ext',
symmetry,
erw closure_eq_iff_is_closed,
exact h,
end | theorem | continuous_map.inf_mem_closed_subalgebra | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"closure_eq_iff_is_closed",
"is_closed",
"set_like.ext'",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_mem_subalgebra_closure (A : subalgebra ℝ C(X, ℝ)) (f g : A) :
(f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A.topological_closure | begin
rw sup_eq,
refine A.topological_closure.smul_mem
(A.topological_closure.add_mem
(A.topological_closure.add_mem (A.le_topological_closure f.property)
(A.le_topological_closure g.property)) _) _,
exact_mod_cast abs_mem_subalgebra_closure A _,
end | theorem | continuous_map.sup_mem_subalgebra_closure | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_mem_closed_subalgebra (A : subalgebra ℝ C(X, ℝ)) (h : is_closed (A : set C(X, ℝ)))
(f g : A) : (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A | begin
convert sup_mem_subalgebra_closure A f g,
apply set_like.ext',
symmetry,
erw closure_eq_iff_is_closed,
exact h,
end | theorem | continuous_map.sup_mem_closed_subalgebra | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"closure_eq_iff_is_closed",
"is_closed",
"set_like.ext'",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sublattice_closure_eq_top
(L : set C(X, ℝ)) (nA : L.nonempty)
(inf_mem : ∀ f g ∈ L, f ⊓ g ∈ L) (sup_mem : ∀ f g ∈ L, f ⊔ g ∈ L)
(sep : L.separates_points_strongly) :
closure L = ⊤ | begin
-- We start by boiling down to a statement about close approximation.
apply eq_top_iff.mpr,
rintros f -,
refine filter.frequently.mem_closure
((filter.has_basis.frequently_iff metric.nhds_basis_ball).mpr (λ ε pos, _)),
simp only [exists_prop, metric.mem_ball],
-- It will be helpful to assume `X` ... | theorem | continuous_map.sublattice_closure_eq_top | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"closure",
"coe_fn_coe_base'",
"compact_space.elim_nhds_subcover",
"continuity",
"continuous_map.inf'_apply",
"exists_prop",
"filter.has_basis.frequently_iff",
"finset",
"finset.inf'_lt_iff",
"finset.inf'_mem",
"finset.lt_inf'_iff",
"finset.lt_sup'_iff",
"finset.sup'_apply",
"finset.sup'_m... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra_topological_closure_eq_top_of_separates_points
(A : subalgebra ℝ C(X, ℝ)) (w : A.separates_points) :
A.topological_closure = ⊤ | begin
-- The closure of `A` is closed under taking `sup` and `inf`,
-- and separates points strongly (since `A` does),
-- so we can apply `sublattice_closure_eq_top`.
apply set_like.ext',
let L := A.topological_closure,
have n : set.nonempty (L : set C(X, ℝ)) :=
⟨(1 : C(X, ℝ)), A.le_topological_closure ... | theorem | continuous_map.subalgebra_topological_closure_eq_top_of_separates_points | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"set.nonempty",
"set_like.ext'",
"subalgebra",
"subalgebra.separates_points.strongly",
"subalgebra.separates_points_monotone"
] | The **Stone-Weierstrass Approximation Theorem**,
that a subalgebra `A` of `C(X, ℝ)`, where `X` is a compact topological space,
is dense if it separates points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map_mem_subalgebra_closure_of_separates_points
(A : subalgebra ℝ C(X, ℝ)) (w : A.separates_points)
(f : C(X, ℝ)) :
f ∈ A.topological_closure | begin
rw subalgebra_topological_closure_eq_top_of_separates_points A w,
simp,
end | theorem | continuous_map.continuous_map_mem_subalgebra_closure_of_separates_points | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"subalgebra"
] | An alternative statement of the Stone-Weierstrass theorem.
If `A` is a subalgebra of `C(X, ℝ)` which separates points (and `X` is compact),
every real-valued continuous function on `X` is a uniform limit of elements of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_mem_subalgebra_near_continuous_map_of_separates_points
(A : subalgebra ℝ C(X, ℝ)) (w : A.separates_points)
(f : C(X, ℝ)) (ε : ℝ) (pos : 0 < ε) :
∃ (g : A), ‖(g : C(X, ℝ)) - f‖ < ε | begin
have w := mem_closure_iff_frequently.mp
(continuous_map_mem_subalgebra_closure_of_separates_points A w f),
rw metric.nhds_basis_ball.frequently_iff at w,
obtain ⟨g, H, m⟩ := w ε pos,
rw [metric.mem_ball, dist_eq_norm] at H,
exact ⟨⟨g, m⟩, H⟩,
end | theorem | continuous_map.exists_mem_subalgebra_near_continuous_map_of_separates_points | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"metric.mem_ball",
"subalgebra"
] | An alternative statement of the Stone-Weierstrass theorem,
for those who like their epsilons.
If `A` is a subalgebra of `C(X, ℝ)` which separates points (and `X` is compact),
every real-valued continuous function on `X` is within any `ε > 0` of some element of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_mem_subalgebra_near_continuous_of_separates_points
(A : subalgebra ℝ C(X, ℝ)) (w : A.separates_points)
(f : X → ℝ) (c : continuous f) (ε : ℝ) (pos : 0 < ε) :
∃ (g : A), ∀ x, ‖g x - f x‖ < ε | begin
obtain ⟨g, b⟩ := exists_mem_subalgebra_near_continuous_map_of_separates_points A w ⟨f, c⟩ ε pos,
use g,
rwa norm_lt_iff _ pos at b,
end | theorem | continuous_map.exists_mem_subalgebra_near_continuous_of_separates_points | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"continuous",
"subalgebra"
] | An alternative statement of the Stone-Weierstrass theorem,
for those who like their epsilons and don't like bundled continuous functions.
If `A` is a subalgebra of `C(X, ℝ)` which separates points (and `X` is compact),
every real-valued continuous function on `X` is within any `ε > 0` of some element of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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