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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lift_cover_restrict' {s : set α} {hs : s ∈ A} :
(lift_cover' A F hF hA).restrict s = F s hs | ext $ lift_cover_coe' | lemma | continuous_map.lift_cover_restrict' | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map (e : α ≃ₜ β) : C(α, β) | ⟨e⟩ | def | homeomorph.to_continuous_map | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | The forward direction of a homeomorphism, as a bundled continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_map_as_coe : f.to_continuous_map = f | rfl | lemma | homeomorph.to_continuous_map_as_coe | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_refl : (homeomorph.refl α : C(α, α)) = continuous_map.id α | rfl | lemma | homeomorph.coe_refl | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous_map.id",
"homeomorph.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_trans : (f.trans g : C(α, γ)) = (g : C(β, γ)).comp f | rfl | lemma | homeomorph.coe_trans | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_comp_to_continuous_map :
(f.symm : C(β, α)).comp (f : C(α, β)) = continuous_map.id α | by rw [← coe_trans, self_trans_symm, coe_refl] | lemma | homeomorph.symm_comp_to_continuous_map | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous_map.id"
] | Left inverse to a continuous map from a homeomorphism, mirroring `equiv.symm_comp_self`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_map_comp_symm :
(f : C(α, β)).comp (f.symm : C(β, α)) = continuous_map.id β | by rw [← coe_trans, symm_trans_self, coe_refl] | lemma | homeomorph.to_continuous_map_comp_symm | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous_map.id"
] | Right inverse to a continuous map from a homeomorphism, mirroring `equiv.self_comp_symm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_continuous_function (α : Type u) (β : Type v)
[topological_space α] [pseudo_metric_space β] extends continuous_map α β :
Type (max u v) | (map_bounded' : ∃ C, ∀ x y, dist (to_fun x) (to_fun y) ≤ C) | structure | bounded_continuous_function | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"continuous_map",
"pseudo_metric_space",
"topological_space"
] | `α →ᵇ β` is the type of bounded continuous functions `α → β` from a topological space to a
metric space.
When possible, instead of parametrizing results over `(f : α →ᵇ β)`,
you should parametrize over `(F : Type*) [bounded_continuous_map_class F α β] (f : F)`.
When you extend this structure, make sure to extend `bou... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_continuous_map_class (F α β : Type*) [topological_space α] [pseudo_metric_space β]
extends continuous_map_class F α β | (map_bounded (f : F) : ∃ C, ∀ x y, dist (f x) (f y) ≤ C) | class | bounded_continuous_map_class | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"continuous_map_class",
"pseudo_metric_space",
"topological_space"
] | `bounded_continuous_map_class F α β` states that `F` is a type of bounded continuous maps.
You should also extend this typeclass when you extend `bounded_continuous_function`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_continuous_fun (f : α →ᵇ β) : (f.to_continuous_map : α → β) = f | rfl | lemma | bounded_continuous_function.coe_to_continuous_fun | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.apply (h : α →ᵇ β) : α → β | h
initialize_simps_projections bounded_continuous_function (to_continuous_map_to_fun → apply) | def | bounded_continuous_function.simps.apply | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"bounded_continuous_function"
] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded (f : α →ᵇ β) : ∃C, ∀ x y : α, dist (f x) (f y) ≤ C | f.map_bounded' | lemma | bounded_continuous_function.bounded | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (f : α →ᵇ β) : continuous f | f.to_continuous_map.continuous | lemma | bounded_continuous_function.continuous | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (h : ∀ x, f x = g x) : f = g | fun_like.ext _ _ h | lemma | bounded_continuous_function.ext | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_range (f : α →ᵇ β) : bounded (range f) | bounded_range_iff.2 f.bounded | lemma | bounded_continuous_function.bounded_range | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_image (f : α →ᵇ β) (s : set α) : bounded (f '' s) | f.bounded_range.mono $ image_subset_range _ _ | lemma | bounded_continuous_function.bounded_image | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_empty [is_empty α] (f g : α →ᵇ β) : f = g | ext $ is_empty.elim ‹_› | lemma | bounded_continuous_function.eq_of_empty | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"is_empty",
"is_empty.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_of_bound (f : C(α, β)) (C : ℝ) (h : ∀ x y : α, dist (f x) (f y) ≤ C) : α →ᵇ β | ⟨f, ⟨C, h⟩⟩ | def | bounded_continuous_function.mk_of_bound | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | A continuous function with an explicit bound is a bounded continuous function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_of_bound_coe {f} {C} {h} : (mk_of_bound f C h : α → β) = (f : α → β) | rfl | lemma | bounded_continuous_function.mk_of_bound_coe | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_of_compact [compact_space α] (f : C(α, β)) : α →ᵇ β | ⟨f, bounded_range_iff.1 (is_compact_range f.continuous).bounded⟩ | def | bounded_continuous_function.mk_of_compact | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"compact_space",
"is_compact_range"
] | A continuous function on a compact space is automatically a bounded continuous function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_of_compact_apply [compact_space α] (f : C(α, β)) (a : α) :
mk_of_compact f a = f a | rfl | lemma | bounded_continuous_function.mk_of_compact_apply | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_of_discrete [discrete_topology α] (f : α → β)
(C : ℝ) (h : ∀ x y : α, dist (f x) (f y) ≤ C) : α →ᵇ β | ⟨⟨f, continuous_of_discrete_topology⟩, ⟨C, h⟩⟩ | def | bounded_continuous_function.mk_of_discrete | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"discrete_topology"
] | If a function is bounded on a discrete space, it is automatically continuous,
and therefore gives rise to an element of the type of bounded continuous functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_eq : dist f g = Inf {C | 0 ≤ C ∧ ∀ x : α, dist (f x) (g x) ≤ C} | rfl | lemma | bounded_continuous_function.dist_eq | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_set_exists : ∃ C, 0 ≤ C ∧ ∀ x : α, dist (f x) (g x) ≤ C | begin
rcases f.bounded_range.union g.bounded_range with ⟨C, hC⟩,
refine ⟨max 0 C, le_max_left _ _, λ x, (hC _ _ _ _).trans (le_max_right _ _)⟩;
[left, right]; apply mem_range_self
end | lemma | bounded_continuous_function.dist_set_exists | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_coe_le_dist (x : α) : dist (f x) (g x) ≤ dist f g | le_cInf dist_set_exists $ λb hb, hb.2 x | lemma | bounded_continuous_function.dist_coe_le_dist | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"le_cInf"
] | The pointwise distance is controlled by the distance between functions, by definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_nonneg' : 0 ≤ dist f g | le_cInf dist_set_exists (λ C, and.left) | lemma | bounded_continuous_function.dist_nonneg' | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"le_cInf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀x:α, dist (f x) (g x) ≤ C | ⟨λ h x, le_trans (dist_coe_le_dist x) h, λ H, cInf_le ⟨0, λ C, and.left⟩ ⟨C0, H⟩⟩ | lemma | bounded_continuous_function.dist_le | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"cInf_le"
] | The distance between two functions is controlled by the supremum of the pointwise distances | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_iff_of_nonempty [nonempty α] :
dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C | ⟨λ h x, le_trans (dist_coe_le_dist x) h,
λ w, (dist_le (le_trans dist_nonneg (w (nonempty.some ‹_›)))).mpr w⟩ | lemma | bounded_continuous_function.dist_le_iff_of_nonempty | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"dist_nonneg",
"nonempty.some"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_lt_of_nonempty_compact [nonempty α] [compact_space α]
(w : ∀x:α, dist (f x) (g x) < C) : dist f g < C | begin
have c : continuous (λ x, dist (f x) (g x)), { continuity, },
obtain ⟨x, -, le⟩ :=
is_compact.exists_forall_ge is_compact_univ set.univ_nonempty (continuous.continuous_on c),
exact lt_of_le_of_lt (dist_le_iff_of_nonempty.mpr (λ y, le y trivial)) (w x),
end | lemma | bounded_continuous_function.dist_lt_of_nonempty_compact | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"compact_space",
"continuity",
"continuous",
"continuous.continuous_on",
"is_compact.exists_forall_ge",
"is_compact_univ",
"set.univ_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_lt_iff_of_compact [compact_space α] (C0 : (0 : ℝ) < C) :
dist f g < C ↔ ∀x:α, dist (f x) (g x) < C | begin
fsplit,
{ intros w x,
exact lt_of_le_of_lt (dist_coe_le_dist x) w, },
{ by_cases h : nonempty α,
{ resetI,
exact dist_lt_of_nonempty_compact, },
{ rintro -,
convert C0,
apply le_antisymm _ dist_nonneg',
rw [dist_eq],
exact cInf_le ⟨0, λ C, and.left⟩ ⟨le_rfl, λ x, fa... | lemma | bounded_continuous_function.dist_lt_iff_of_compact | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"cInf_le",
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_lt_iff_of_nonempty_compact [nonempty α] [compact_space α] :
dist f g < C ↔ ∀x:α, dist (f x) (g x) < C | ⟨λ w x, lt_of_le_of_lt (dist_coe_le_dist x) w, dist_lt_of_nonempty_compact⟩ | lemma | bounded_continuous_function.dist_lt_iff_of_nonempty_compact | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_eq : nndist f g = Inf {C | ∀ x : α, nndist (f x) (g x) ≤ C} | subtype.ext $ dist_eq.trans $ begin
rw [nnreal.coe_Inf, nnreal.coe_image],
simp_rw [mem_set_of_eq, ←nnreal.coe_le_coe, subtype.coe_mk, exists_prop, coe_nndist],
end | lemma | bounded_continuous_function.nndist_eq | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"coe_nndist",
"exists_prop",
"nnreal.coe_Inf",
"nnreal.coe_image",
"subtype.coe_mk",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_set_exists : ∃ C, ∀ x : α, nndist (f x) (g x) ≤ C | subtype.exists.mpr $ dist_set_exists.imp $ λ a ⟨ha, h⟩, ⟨ha, h⟩ | lemma | bounded_continuous_function.nndist_set_exists | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_coe_le_nndist (x : α) : nndist (f x) (g x) ≤ nndist f g | dist_coe_le_dist x | lemma | bounded_continuous_function.nndist_coe_le_nndist | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_zero_of_empty [is_empty α] : dist f g = 0 | by rw [(ext is_empty_elim : f = g), dist_self] | lemma | bounded_continuous_function.dist_zero_of_empty | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"dist_self",
"is_empty",
"is_empty_elim"
] | On an empty space, bounded continuous functions are at distance 0 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_eq_supr : dist f g = ⨆ x : α, dist (f x) (g x) | begin
casesI is_empty_or_nonempty α, { rw [supr_of_empty', real.Sup_empty, dist_zero_of_empty] },
refine (dist_le_iff_of_nonempty.mpr $ le_csupr _).antisymm (csupr_le dist_coe_le_dist),
exact dist_set_exists.imp (λ C hC, forall_range_iff.2 hC.2)
end | lemma | bounded_continuous_function.dist_eq_supr | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"csupr_le",
"is_empty_or_nonempty",
"le_csupr",
"real.Sup_empty",
"supr_of_empty'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_eq_supr : nndist f g = ⨆ x : α, nndist (f x) (g x) | subtype.ext $ dist_eq_supr.trans $ by simp_rw [nnreal.coe_supr, coe_nndist] | lemma | bounded_continuous_function.nndist_eq_supr | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"coe_nndist",
"nnreal.coe_supr",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iff_tendsto_uniformly {ι : Type*} {F : ι → (α →ᵇ β)} {f : α →ᵇ β} {l : filter ι} :
tendsto F l (𝓝 f) ↔ tendsto_uniformly (λ i, F i) f l | iff.intro
(λ h, tendsto_uniformly_iff.2
(λ ε ε0, (metric.tendsto_nhds.mp h ε ε0).mp (eventually_of_forall $
λ n hn x, lt_of_le_of_lt (dist_coe_le_dist x) (dist_comm (F n) f ▸ hn))))
(λ h, metric.tendsto_nhds.mpr $ λ ε ε_pos,
(h _ (dist_mem_uniformity $ half_pos ε_pos)).mp (eventually_of_forall $
λ n... | lemma | bounded_continuous_function.tendsto_iff_tendsto_uniformly | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"dist_comm",
"filter",
"half_pos",
"tendsto_uniformly"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_coe_fn : inducing (uniform_fun.of_fun ∘ coe_fn : (α →ᵇ β) → (α →ᵤ β)) | begin
rw inducing_iff_nhds,
refine λ f, eq_of_forall_le_iff (λ l, _),
rw [← tendsto_iff_comap, ← tendsto_id', tendsto_iff_tendsto_uniformly,
uniform_fun.tendsto_iff_tendsto_uniformly],
refl
end | lemma | bounded_continuous_function.inducing_coe_fn | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"eq_of_forall_le_iff",
"inducing",
"inducing_iff_nhds",
"uniform_fun.of_fun",
"uniform_fun.tendsto_iff_tendsto_uniformly"
] | The topology on `α →ᵇ β` is exactly the topology induced by the natural map to `α →ᵤ β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding_coe_fn : embedding (uniform_fun.of_fun ∘ coe_fn : (α →ᵇ β) → (α →ᵤ β)) | ⟨inducing_coe_fn, λ f g h, ext $ λ x, congr_fun h x⟩ | lemma | bounded_continuous_function.embedding_coe_fn | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"embedding",
"uniform_fun.of_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (b : β) : α →ᵇ β | ⟨continuous_map.const α b, 0, by simp [le_rfl]⟩ | def | bounded_continuous_function.const | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"le_rfl"
] | Constant as a continuous bounded function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_apply' (a : α) (b : β) : (const α b : α → β) a = b | rfl | lemma | bounded_continuous_function.const_apply' | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_evalx (x : α) : lipschitz_with 1 (λ f : α →ᵇ β, f x) | lipschitz_with.mk_one $ λ f g, dist_coe_le_dist x | lemma | bounded_continuous_function.lipschitz_evalx | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"lipschitz_with",
"lipschitz_with.mk_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_coe : @uniform_continuous (α →ᵇ β) (α → β) _ _ coe_fn | uniform_continuous_pi.2 $ λ x, (lipschitz_evalx x).uniform_continuous | theorem | bounded_continuous_function.uniform_continuous_coe | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe : continuous (λ (f : α →ᵇ β) x, f x) | uniform_continuous.continuous uniform_continuous_coe | lemma | bounded_continuous_function.continuous_coe | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"continuous",
"uniform_continuous.continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_eval_const {x : α} : continuous (λ f : α →ᵇ β, f x) | (continuous_apply x).comp continuous_coe | theorem | bounded_continuous_function.continuous_eval_const | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"continuous",
"continuous_apply"
] | When `x` is fixed, `(f : α →ᵇ β) ↦ f x` is continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_eval : continuous (λ p : (α →ᵇ β) × α, p.1 p.2) | continuous_prod_of_continuous_lipschitz _ 1 (λ f, f.continuous) $ lipschitz_evalx | theorem | bounded_continuous_function.continuous_eval | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"continuous",
"continuous_prod_of_continuous_lipschitz"
] | The evaluation map is continuous, as a joint function of `u` and `x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous {δ : Type*} [topological_space δ] (f : α →ᵇ β) (g : C(δ, α)) : δ →ᵇ β | { to_continuous_map := f.1.comp g,
map_bounded' := f.map_bounded'.imp (λ C hC x y, hC _ _) } | def | bounded_continuous_function.comp_continuous | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"topological_space"
] | Composition of a bounded continuous function and a continuous function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp_continuous {δ : Type*} [topological_space δ] (f : α →ᵇ β) (g : C(δ, α)) :
coe_fn (f.comp_continuous g) = f ∘ g | rfl | lemma | bounded_continuous_function.coe_comp_continuous | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_apply {δ : Type*} [topological_space δ]
(f : α →ᵇ β) (g : C(δ, α)) (x : δ) : f.comp_continuous g x = f (g x) | rfl | lemma | bounded_continuous_function.comp_continuous_apply | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_comp_continuous {δ : Type*} [topological_space δ] (g : C(δ, α)) :
lipschitz_with 1 (λ f : α →ᵇ β, f.comp_continuous g) | lipschitz_with.mk_one $ λ f₁ f₂, (dist_le dist_nonneg).2 $ λ x, dist_coe_le_dist (g x) | lemma | bounded_continuous_function.lipschitz_comp_continuous | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"dist_nonneg",
"lipschitz_with",
"lipschitz_with.mk_one",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_comp_continuous {δ : Type*} [topological_space δ] (g : C(δ, α)) :
continuous (λ f : α →ᵇ β, f.comp_continuous g) | (lipschitz_comp_continuous g).continuous | lemma | bounded_continuous_function.continuous_comp_continuous | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict (f : α →ᵇ β) (s : set α) : s →ᵇ β | f.comp_continuous $ (continuous_map.id _).restrict s | def | bounded_continuous_function.restrict | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"continuous_map.id"
] | Restrict a bounded continuous function to a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_restrict (f : α →ᵇ β) (s : set α) : coe_fn (f.restrict s) = f ∘ coe | rfl | lemma | bounded_continuous_function.coe_restrict | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_apply (f : α →ᵇ β) (s : set α) (x : s) : f.restrict s x = f x | rfl | lemma | bounded_continuous_function.restrict_apply | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (G : β → γ) {C : ℝ≥0} (H : lipschitz_with C G)
(f : α →ᵇ β) : α →ᵇ γ | ⟨⟨λx, G (f x), H.continuous.comp f.continuous⟩,
let ⟨D, hD⟩ := f.bounded in
⟨max C 0 * D, λ x y, calc
dist (G (f x)) (G (f y)) ≤ C * dist (f x) (f y) : H.dist_le_mul _ _
... ≤ max C 0 * dist (f x) (f y) : mul_le_mul_of_nonneg_right (le_max_left C 0) dist_nonneg
... ≤ max C 0 * D : mul_le_mul_of_nonneg_l... | def | bounded_continuous_function.comp | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"dist_nonneg",
"lipschitz_with",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right"
] | Composition (in the target) of a bounded continuous function with a Lipschitz map again
gives a bounded continuous function | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_comp {G : β → γ} {C : ℝ≥0} (H : lipschitz_with C G) :
lipschitz_with C (comp G H : (α →ᵇ β) → α →ᵇ γ) | lipschitz_with.of_dist_le_mul $ λ f g,
(dist_le (mul_nonneg C.2 dist_nonneg)).2 $ λ x,
calc dist (G (f x)) (G (g x)) ≤ C * dist (f x) (g x) : H.dist_le_mul _ _
... ≤ C * dist f g : mul_le_mul_of_nonneg_left (dist_coe_le_dist _) C.2 | lemma | bounded_continuous_function.lipschitz_comp | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"dist_nonneg",
"lipschitz_with",
"mul_le_mul_of_nonneg_left"
] | The composition operator (in the target) with a Lipschitz map is Lipschitz | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_comp {G : β → γ} {C : ℝ≥0} (H : lipschitz_with C G) :
uniform_continuous (comp G H : (α →ᵇ β) → α →ᵇ γ) | (lipschitz_comp H).uniform_continuous | lemma | bounded_continuous_function.uniform_continuous_comp | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"lipschitz_with",
"uniform_continuous"
] | The composition operator (in the target) with a Lipschitz map is uniformly continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_comp {G : β → γ} {C : ℝ≥0} (H : lipschitz_with C G) :
continuous (comp G H : (α →ᵇ β) → α →ᵇ γ) | (lipschitz_comp H).continuous | lemma | bounded_continuous_function.continuous_comp | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"continuous",
"lipschitz_with"
] | The composition operator (in the target) with a Lipschitz map is continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cod_restrict (s : set β) (f : α →ᵇ β) (H : ∀x, f x ∈ s) : α →ᵇ s | ⟨⟨s.cod_restrict f H, f.continuous.subtype_mk _⟩, f.bounded⟩ | def | bounded_continuous_function.cod_restrict | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | Restriction (in the target) of a bounded continuous function taking values in a subset | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend (f : α ↪ δ) (g : α →ᵇ β) (h : δ →ᵇ β) : δ →ᵇ β | { to_fun := extend f g h,
continuous_to_fun := continuous_of_discrete_topology,
map_bounded' :=
begin
rw [← bounded_range_iff, range_extend f.injective, metric.bounded_union],
exact ⟨g.bounded_range, h.bounded_image _⟩
end } | def | bounded_continuous_function.extend | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"continuous_of_discrete_topology",
"extend",
"metric.bounded_union"
] | A version of `function.extend` for bounded continuous maps. We assume that the domain has
discrete topology, so we only need to verify boundedness. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_apply (f : α ↪ δ) (g : α →ᵇ β) (h : δ →ᵇ β) (x : α) :
extend f g h (f x) = g x | f.injective.extend_apply _ _ _ | lemma | bounded_continuous_function.extend_apply | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"extend"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_comp (f : α ↪ δ) (g : α →ᵇ β) (h : δ →ᵇ β) : extend f g h ∘ f = g | extend_comp f.injective _ _ | lemma | bounded_continuous_function.extend_comp | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"extend",
"extend_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_apply' {f : α ↪ δ} {x : δ} (hx : x ∉ range f) (g : α →ᵇ β) (h : δ →ᵇ β) :
extend f g h x = h x | extend_apply' _ _ _ hx | lemma | bounded_continuous_function.extend_apply' | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"extend",
"extend_apply'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_of_empty [is_empty α] (f : α ↪ δ) (g : α →ᵇ β) (h : δ →ᵇ β) :
extend f g h = h | fun_like.coe_injective $ function.extend_of_empty f g h | lemma | bounded_continuous_function.extend_of_empty | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"extend",
"fun_like.coe_injective",
"function.extend_of_empty",
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_extend_extend (f : α ↪ δ) (g₁ g₂ : α →ᵇ β) (h₁ h₂ : δ →ᵇ β) :
dist (g₁.extend f h₁) (g₂.extend f h₂) =
max (dist g₁ g₂) (dist (h₁.restrict (range f)ᶜ) (h₂.restrict (range f)ᶜ)) | begin
refine le_antisymm ((dist_le $ le_max_iff.2 $ or.inl dist_nonneg).2 $ λ x, _) (max_le _ _),
{ rcases em (∃ y, f y = x) with (⟨x, rfl⟩|hx),
{ simp only [extend_apply],
exact (dist_coe_le_dist x).trans (le_max_left _ _) },
{ simp only [extend_apply' hx],
lift x to ((range f)ᶜ : set δ) using ... | lemma | bounded_continuous_function.dist_extend_extend | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"dist_nonneg",
"em",
"extend",
"extend_apply'",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_extend (f : α ↪ δ) (h : δ →ᵇ β) :
isometry (λ g : α →ᵇ β, extend f g h) | isometry.of_dist_eq $ λ g₁ g₂, by simp [dist_nonneg] | lemma | bounded_continuous_function.isometry_extend | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"dist_nonneg",
"extend",
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arzela_ascoli₁ [compact_space β]
(A : set (α →ᵇ β))
(closed : is_closed A)
(H : equicontinuous (coe_fn : A → α → β)) :
is_compact A | begin
simp_rw [equicontinuous, metric.equicontinuous_at_iff_pair] at H,
refine is_compact_of_totally_bounded_is_closed _ closed,
refine totally_bounded_of_finite_discretization (λ ε ε0, _),
rcases exists_between ε0 with ⟨ε₁, ε₁0, εε₁⟩,
let ε₂ := ε₁/2/2,
/- We have to find a finite discretization of `u`, i.e... | theorem | bounded_continuous_function.arzela_ascoli₁ | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"add_halves",
"compact_space",
"dist_triangle4_right",
"dist_triangle_right",
"equicontinuous",
"exists_between",
"finite_cover_balls_of_compact",
"half_pos",
"is_closed",
"is_compact",
"is_compact_of_totally_bounded_is_closed",
"is_compact_univ",
"is_open",
"metric.equicontinuous_at_iff_p... | First version, with pointwise equicontinuity and range in a compact space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arzela_ascoli₂
(s : set β) (hs : is_compact s)
(A : set (α →ᵇ β))
(closed : is_closed A)
(in_s : ∀(f : α →ᵇ β) (x : α), f ∈ A → f x ∈ s)
(H : equicontinuous (coe_fn : A → α → β)) :
is_compact A | /- This version is deduced from the previous one by restricting to the compact type in the target,
using compactness there and then lifting everything to the original space. -/
begin
have M : lipschitz_with 1 coe := lipschitz_with.subtype_coe s,
let F : (α →ᵇ s) → α →ᵇ β := comp coe M,
refine is_compact_of_is_clo... | theorem | bounded_continuous_function.arzela_ascoli₂ | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"compact_space",
"equicontinuous",
"is_closed",
"is_compact",
"is_compact_of_is_closed_subset",
"lipschitz_with",
"lipschitz_with.subtype_coe"
] | Second version, with pointwise equicontinuity and range in a compact subset | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arzela_ascoli [t2_space β]
(s : set β) (hs : is_compact s)
(A : set (α →ᵇ β))
(in_s : ∀(f : α →ᵇ β) (x : α), f ∈ A → f x ∈ s)
(H : equicontinuous (coe_fn : A → α → β)) :
is_compact (closure A) | /- This version is deduced from the previous one by checking that the closure of A, in
addition to being closed, still satisfies the properties of compact range and equicontinuity -/
arzela_ascoli₂ s hs (closure A) is_closed_closure
(λ f x hf, (mem_of_closed' hs.is_closed).2 $ λ ε ε0,
let ⟨g, gA, dist_fg⟩ := metr... | theorem | bounded_continuous_function.arzela_ascoli | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"closure",
"equicontinuous",
"is_closed_closure",
"is_compact",
"t2_space"
] | Third (main) version, with pointwise equicontinuity and range in a compact subset, but
without closedness. The closure is then compact | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_one : ((1 : α →ᵇ β) : α → β) = 1 | rfl | lemma | bounded_continuous_function.coe_one | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_of_compact_one [compact_space α] : mk_of_compact (1 : C(α, β)) = 1 | rfl | lemma | bounded_continuous_function.mk_of_compact_one | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forall_coe_one_iff_one (f : α →ᵇ β) : (∀ x, f x = 1) ↔ f = 1 | (@fun_like.ext_iff _ _ _ _ f 1).symm | lemma | bounded_continuous_function.forall_coe_one_iff_one | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_comp_continuous [topological_space γ] (f : C(γ, α)) :
(1 : α →ᵇ β).comp_continuous f = 1 | rfl | lemma | bounded_continuous_function.one_comp_continuous | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add : ⇑(f + g) = f + g | rfl | lemma | bounded_continuous_function.coe_add | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply : (f + g) x = f x + g x | rfl | lemma | bounded_continuous_function.add_apply | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_of_compact_add [compact_space α] (f g : C(α, β)) :
mk_of_compact (f + g) = mk_of_compact f + mk_of_compact g | rfl | lemma | bounded_continuous_function.mk_of_compact_add | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comp_continuous [topological_space γ] (h : C(γ, α)) :
(g + f).comp_continuous h = g.comp_continuous h + f.comp_continuous h | rfl | lemma | bounded_continuous_function.add_comp_continuous | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nsmul_rec : ∀ n, ⇑(nsmul_rec n f) = n • f | | 0 := by rw [nsmul_rec, zero_smul, coe_zero]
| (n + 1) := by rw [nsmul_rec, succ_nsmul, coe_add, coe_nsmul_rec] | lemma | bounded_continuous_function.coe_nsmul_rec | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"nsmul_rec",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nat_scalar : has_smul ℕ (α →ᵇ β) | { smul := λ n f,
{ to_continuous_map := n • f.to_continuous_map,
map_bounded' := by simpa [coe_nsmul_rec] using (nsmul_rec n f).map_bounded' } } | instance | bounded_continuous_function.has_nat_scalar | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"has_smul",
"nsmul_rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nsmul (r : ℕ) (f : α →ᵇ β) : ⇑(r • f) = r • f | rfl | lemma | bounded_continuous_function.coe_nsmul | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_apply (r : ℕ) (f : α →ᵇ β) (v : α) : (r • f) v = r • f v | rfl | lemma | bounded_continuous_function.nsmul_apply | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_add_hom : (α →ᵇ β) →+ (α → β) | { to_fun := coe_fn, map_zero' := coe_zero, map_add' := coe_add } | def | bounded_continuous_function.coe_fn_add_hom | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | Coercion of a `normed_add_group_hom` is an `add_monoid_hom`. Similar to
`add_monoid_hom.coe_fn`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_map_add_hom : (α →ᵇ β) →+ C(α, β) | { to_fun := to_continuous_map,
map_zero' := by { ext, simp, },
map_add' := by { intros, ext, simp, }, } | def | bounded_continuous_function.to_continuous_map_add_hom | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | The additive map forgetting that a bounded continuous function is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_sum {ι : Type*} (s : finset ι) (f : ι → (α →ᵇ β)) :
⇑(∑ i in s, f i) = (∑ i in s, (f i : α → β)) | (@coe_fn_add_hom α β _ _ _ _).map_sum f s | lemma | bounded_continuous_function.coe_sum | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_apply {ι : Type*} (s : finset ι) (f : ι → (α →ᵇ β)) (a : α) :
(∑ i in s, f i) a = (∑ i in s, f i a) | by simp | lemma | bounded_continuous_function.sum_apply | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_def : ‖f‖ = dist f 0 | rfl | lemma | bounded_continuous_function.norm_def | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq (f : α →ᵇ β) :
‖f‖ = Inf {C : ℝ | 0 ≤ C ∧ ∀ (x : α), ‖f x‖ ≤ C} | by simp [norm_def, bounded_continuous_function.dist_eq] | lemma | bounded_continuous_function.norm_eq | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"bounded_continuous_function.dist_eq"
] | The norm of a bounded continuous function is the supremum of `‖f x‖`.
We use `Inf` to ensure that the definition works if `α` has no elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_eq_of_nonempty [h : nonempty α] : ‖f‖ = Inf {C : ℝ | ∀ (x : α), ‖f x‖ ≤ C} | begin
unfreezingI { obtain ⟨a⟩ := h, },
rw norm_eq,
congr,
ext,
simp only [and_iff_right_iff_imp],
exact λ h', le_trans (norm_nonneg (f a)) (h' a),
end | lemma | bounded_continuous_function.norm_eq_of_nonempty | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"and_iff_right_iff_imp"
] | When the domain is non-empty, we do not need the `0 ≤ C` condition in the formula for ‖f‖ as an
`Inf`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_eq_zero_of_empty [h : is_empty α] : ‖f‖ = 0 | dist_zero_of_empty | lemma | bounded_continuous_function.norm_eq_zero_of_empty | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ | calc
‖f x‖ = dist (f x) ((0 : α →ᵇ β) x) : by simp [dist_zero_right]
... ≤ ‖f‖ : dist_coe_le_dist _ | lemma | bounded_continuous_function.norm_coe_le_norm | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_two_norm' {f : γ → β} {C : ℝ} (hC : ∀ x, ‖f x‖ ≤ C) (x y : γ) :
dist (f x) (f y) ≤ 2 * C | calc dist (f x) (f y) ≤ ‖f x‖ + ‖f y‖ : dist_le_norm_add_norm _ _
... ≤ C + C : add_le_add (hC x) (hC y)
... = 2 * C : (two_mul _).symm | lemma | bounded_continuous_function.dist_le_two_norm' | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ | dist_le_two_norm' f.norm_coe_le_norm x y | lemma | bounded_continuous_function.dist_le_two_norm | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | Distance between the images of any two points is at most twice the norm of the function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_le (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀x:α, ‖f x‖ ≤ C | by simpa using @dist_le _ _ _ _ f 0 _ C0 | lemma | bounded_continuous_function.norm_le | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | The norm of a function is controlled by the supremum of the pointwise norms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_le_of_nonempty [nonempty α]
{f : α →ᵇ β} {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M | begin
simp_rw [norm_def, ←dist_zero_right],
exact dist_le_iff_of_nonempty,
end | lemma | bounded_continuous_function.norm_le_of_nonempty | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_lt_iff_of_compact [compact_space α]
{f : α →ᵇ β} {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M | begin
simp_rw [norm_def, ←dist_zero_right],
exact dist_lt_iff_of_compact M0,
end | lemma | bounded_continuous_function.norm_lt_iff_of_compact | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_lt_iff_of_nonempty_compact [nonempty α] [compact_space α]
{f : α →ᵇ β} {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M | begin
simp_rw [norm_def, ←dist_zero_right],
exact dist_lt_iff_of_nonempty_compact,
end | lemma | bounded_continuous_function.norm_lt_iff_of_nonempty_compact | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_const_le (b : β) : ‖const α b‖ ≤ ‖b‖ | (norm_le (norm_nonneg b)).2 $ λ x, le_rfl | lemma | bounded_continuous_function.norm_const_le | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"le_rfl"
] | Norm of `const α b` is less than or equal to `‖b‖`. If `α` is nonempty,
then it is equal to `‖b‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_const_eq [h : nonempty α] (b : β) : ‖const α b‖ = ‖b‖ | le_antisymm (norm_const_le b) $ h.elim $ λ x, (const α b).norm_coe_le_norm x | lemma | bounded_continuous_function.norm_const_eq | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_normed_add_comm_group {α : Type u} {β : Type v} [topological_space α]
[seminormed_add_comm_group β] (f : α → β) (Hf : continuous f) (C : ℝ) (H : ∀x, ‖f x‖ ≤ C) :
α →ᵇ β | ⟨⟨λn, f n, Hf⟩, ⟨_, dist_le_two_norm' H⟩⟩ | def | bounded_continuous_function.of_normed_add_comm_group | topology.continuous_function | src/topology/continuous_function/bounded.lean | [
"analysis.normed.order.lattice",
"analysis.normed_space.operator_norm",
"analysis.normed_space.star.basic",
"data.real.sqrt",
"topology.continuous_function.algebra",
"topology.metric_space.equicontinuity"
] | [
"continuous",
"seminormed_add_comm_group",
"topological_space"
] | Constructing a bounded continuous function from a uniformly bounded continuous
function taking values in a normed group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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