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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coe_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) :
(of_normed_add_comm_group f Hf C H : α → β) = f | rfl | lemma | bounded_continuous_function.coe_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_of_normed_add_comm_group_le {f : α → β} (hfc : continuous f) {C : ℝ} (hC : 0 ≤ C)
(hfC : ∀ x, ‖f x‖ ≤ C) : ‖of_normed_add_comm_group f hfc C hfC‖ ≤ C | (norm_le hC).2 hfC | lemma | bounded_continuous_function.norm_of_normed_add_comm_group_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"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_normed_add_comm_group_discrete {α : Type u} {β : Type v}
[topological_space α] [discrete_topology α] [seminormed_add_comm_group β]
(f : α → β) (C : ℝ) (H : ∀x, norm (f x) ≤ C) : α →ᵇ β | of_normed_add_comm_group f continuous_of_discrete_topology C H | def | bounded_continuous_function.of_normed_add_comm_group_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"
] | [
"continuous_of_discrete_topology",
"discrete_topology",
"seminormed_add_comm_group",
"topological_space"
] | Constructing a bounded continuous function from a uniformly bounded
function on a discrete space, taking values in a normed group | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_normed_add_comm_group_discrete {α : Type u} {β : Type v} [topological_space α]
[discrete_topology α] [seminormed_add_comm_group β] (f : α → β) (C : ℝ) (H : ∀x, ‖f x‖ ≤ C) :
(of_normed_add_comm_group_discrete f C H : α → β) = f | rfl | lemma | bounded_continuous_function.coe_of_normed_add_comm_group_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",
"seminormed_add_comm_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_comp : α →ᵇ ℝ | f.comp norm lipschitz_with_one_norm | def | bounded_continuous_function.norm_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"
] | [] | Taking the pointwise norm of a bounded continuous function with values in a
`seminormed_add_comm_group` yields a bounded continuous function with values in ℝ. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_norm_comp : (f.norm_comp : α → ℝ) = norm ∘ f | rfl | lemma | bounded_continuous_function.coe_norm_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_norm_comp : ‖f.norm_comp‖ = ‖f‖ | by simp only [norm_eq, coe_norm_comp, norm_norm] | lemma | bounded_continuous_function.norm_norm_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"
] | [
"norm_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_range_norm_comp : bdd_above $ set.range $ norm ∘ f | (real.bounded_iff_bdd_below_bdd_above.mp $ @bounded_range _ _ _ _ f.norm_comp).2 | lemma | bounded_continuous_function.bdd_above_range_norm_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"
] | [
"bdd_above",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_supr_norm : ‖f‖ = ⨆ x : α, ‖f x‖ | by simp_rw [norm_def, dist_eq_supr, coe_zero, pi.zero_apply, dist_zero_right] | lemma | bounded_continuous_function.norm_eq_supr_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 | |
coe_neg : ⇑(-f) = -f | rfl | lemma | bounded_continuous_function.coe_neg | 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 | |
neg_apply : (-f) x = -f x | rfl | lemma | bounded_continuous_function.neg_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_sub : ⇑(f - g) = f - g | rfl | lemma | bounded_continuous_function.coe_sub | 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 | |
sub_apply : (f - g) x = f x - g x | rfl | lemma | bounded_continuous_function.sub_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_neg [compact_space α] (f : C(α, β)) :
mk_of_compact (-f) = -mk_of_compact f | rfl | lemma | bounded_continuous_function.mk_of_compact_neg | 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_compact_sub [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_sub | 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 | |
coe_zsmul_rec : ∀ z, ⇑(zsmul_rec z f) = z • f | | (int.of_nat n) := by rw [zsmul_rec, int.of_nat_eq_coe, coe_nsmul_rec, coe_nat_zsmul]
| -[1+ n] := by rw [zsmul_rec, zsmul_neg_succ_of_nat, coe_neg, coe_nsmul_rec] | lemma | bounded_continuous_function.coe_zsmul_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"
] | [
"zsmul_rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_int_scalar : has_smul ℤ (α →ᵇ β) | { smul := λ n f,
{ to_continuous_map := n • f.to_continuous_map,
map_bounded' := by simpa using (zsmul_rec n f).map_bounded' } } | instance | bounded_continuous_function.has_int_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",
"zsmul_rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zsmul (r : ℤ) (f : α →ᵇ β) : ⇑(r • f) = r • f | rfl | lemma | bounded_continuous_function.coe_zsmul | 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 | |
zsmul_apply (r : ℤ) (f : α →ᵇ β) (v : α) : (r • f) v = r • f v | rfl | lemma | bounded_continuous_function.zsmul_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 | |
nnnorm_def : ‖f‖₊ = nndist f 0 | rfl | lemma | bounded_continuous_function.nnnorm_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 | |
nnnorm_coe_le_nnnorm (x : α) : ‖f x‖₊ ≤ ‖f‖₊ | norm_coe_le_norm _ _ | lemma | bounded_continuous_function.nnnorm_coe_le_nnnorm | 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_le_two_nnnorm (x y : α) : nndist (f x) (f y) ≤ 2 * ‖f‖₊ | dist_le_two_norm _ _ _ | lemma | bounded_continuous_function.nndist_le_two_nnnorm | 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 | |
nnnorm_le (C : ℝ≥0) : ‖f‖₊ ≤ C ↔ ∀x:α, ‖f x‖₊ ≤ C | norm_le C.prop | lemma | bounded_continuous_function.nnnorm_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 nnnorm of a function is controlled by the supremum of the pointwise nnnorms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_const_le (b : β) : ‖const α b‖₊ ≤ ‖b‖₊ | norm_const_le _ | lemma | bounded_continuous_function.nnnorm_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_const_eq [h : nonempty α] (b : β) : ‖const α b‖₊ = ‖b‖₊ | subtype.ext $ norm_const_eq _ | lemma | bounded_continuous_function.nnnorm_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"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_eq_supr_nnnorm : ‖f‖₊ = ⨆ x : α, ‖f x‖₊ | subtype.ext $ (norm_eq_supr_norm f).trans $ by simp_rw [nnreal.coe_supr, coe_nnnorm] | lemma | bounded_continuous_function.nnnorm_eq_supr_nnnorm | 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"
] | [
"nnreal.coe_supr",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_diff_coe_le_dist : ‖f x - g x‖ ≤ dist f g | by { rw dist_eq_norm, exact (f - g).norm_coe_le_norm x } | lemma | bounded_continuous_function.abs_diff_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le_coe_add_dist {f g : α →ᵇ ℝ} : f x ≤ g x + dist f g | sub_le_iff_le_add'.1 $ (abs_le.1 $ @dist_coe_le_dist _ _ _ _ f g x).2 | lemma | bounded_continuous_function.coe_le_coe_add_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_comp_continuous_le [topological_space γ] (f : α →ᵇ β) (g : C(γ, α)) :
‖f.comp_continuous g‖ ≤ ‖f‖ | ((lipschitz_comp_continuous g).dist_le_mul f 0).trans $
by rw [nnreal.coe_one, one_mul, dist_zero_right] | lemma | bounded_continuous_function.norm_comp_continuous_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"
] | [
"nnreal.coe_one",
"one_mul",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (c : 𝕜) (f : α →ᵇ β) : ⇑(c • f) = λ x, c • (f x) | rfl | lemma | bounded_continuous_function.coe_smul | 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 | |
smul_apply (c : 𝕜) (f : α →ᵇ β) (x : α) : (c • f) x = c • f x | rfl | lemma | bounded_continuous_function.smul_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 | |
eval_clm (x : α) : (α →ᵇ β) →L[𝕜] β | { to_fun := λ f, f x,
map_add' := λ f g, add_apply _ _,
map_smul' := λ c f, smul_apply _ _ _ } | def | bounded_continuous_function.eval_clm | 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 evaluation at a point, as a continuous linear map from `α →ᵇ β` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_clm_apply (x : α) (f : α →ᵇ β) :
eval_clm 𝕜 x f = f x | rfl | lemma | bounded_continuous_function.eval_clm_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 | |
to_continuous_map_linear_map : (α →ᵇ β) →ₗ[𝕜] C(α, β) | { to_fun := to_continuous_map,
map_smul' := λ f g, rfl,
map_add' := λ c f, rfl } | def | bounded_continuous_function.to_continuous_map_linear_map | 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 linear map forgetting that a bounded continuous function is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.continuous_linear_map.comp_left_continuous_bounded (g : β →L[𝕜] γ) :
(α →ᵇ β) →L[𝕜] (α →ᵇ γ) | linear_map.mk_continuous
{ to_fun := λ f, of_normed_add_comm_group
(g ∘ f)
(g.continuous.comp f.continuous)
(‖g‖ * ‖f‖)
(λ x, (g.le_op_norm_of_le (f.norm_coe_le_norm x))),
map_add' := λ f g, by ext; simp,
map_smul' := λ c f, by ext; simp }
‖g‖
(λ f, norm_of_normed_add_comm_group_le... | def | continuous_linear_map.comp_left_continuous_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"
] | [
"linear_map.mk_continuous"
] | Postcomposition of bounded continuous functions into a normed module by a continuous linear map is
a continuous linear map.
Upgraded version of `continuous_linear_map.comp_left_continuous`, similar to
`linear_map.comp_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.continuous_linear_map.comp_left_continuous_bounded_apply (g : β →L[𝕜] γ)
(f : α →ᵇ β) (x : α) :
(g.comp_left_continuous_bounded α f) x = g (f x) | rfl | lemma | continuous_linear_map.comp_left_continuous_bounded_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_mul (f g : α →ᵇ R) : ⇑(f * g) = f * g | rfl | lemma | bounded_continuous_function.coe_mul | 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 | |
mul_apply (f g : α →ᵇ R) (x : α) : (f * g) x = f x * g x | rfl | lemma | bounded_continuous_function.mul_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_npow_rec (f : α →ᵇ R) : ∀ n, ⇑(npow_rec n f) = f ^ n | | 0 := by rw [npow_rec, pow_zero, coe_one]
| (n + 1) := by rw [npow_rec, pow_succ, coe_mul, coe_npow_rec] | lemma | bounded_continuous_function.coe_npow_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"
] | [
"npow_rec",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nat_pow : has_pow (α →ᵇ R) ℕ | { pow := λ f n,
{ to_continuous_map := f.to_continuous_map ^ n,
map_bounded' := by simpa [coe_npow_rec] using (npow_rec n f).map_bounded' } } | instance | bounded_continuous_function.has_nat_pow | 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"
] | [
"npow_rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (n : ℕ) (f : α →ᵇ R) : ⇑(f ^ n) = f ^ n | rfl | lemma | bounded_continuous_function.coe_pow | 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 | |
pow_apply (n : ℕ) (f : α →ᵇ R) (v : α) : (f ^ n) v = f v ^ n | rfl | lemma | bounded_continuous_function.pow_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_nat_cast (n : ℕ) : ((n : α →ᵇ R) : α → R) = n | rfl | lemma | bounded_continuous_function.coe_nat_cast | 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_int_cast (n : ℤ) : ((n : α →ᵇ R) : α → R) = n | rfl | lemma | bounded_continuous_function.coe_int_cast | 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 | |
C : 𝕜 →+* (α →ᵇ γ) | { to_fun := λ (c : 𝕜), const α ((algebra_map 𝕜 γ) c),
map_one' := ext $ λ x, (algebra_map 𝕜 γ).map_one,
map_mul' := λ c₁ c₂, ext $ λ x, (algebra_map 𝕜 γ).map_mul _ _,
map_zero' := ext $ λ x, (algebra_map 𝕜 γ).map_zero,
map_add' := λ c₁ c₂, ext $ λ x, (algebra_map 𝕜 γ).map_add _ _ } | def | bounded_continuous_function.C | 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"
] | [
"algebra_map",
"map_mul",
"map_one"
] | `bounded_continuous_function.const` as a `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_apply (k : 𝕜) (a : α) :
algebra_map 𝕜 (α →ᵇ γ) k a = k • 1 | by { rw algebra.algebra_map_eq_smul_one, refl, } | lemma | bounded_continuous_function.algebra_map_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"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"algebra_map_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_smul' : has_smul (α →ᵇ 𝕜) (α →ᵇ β) | ⟨λ (f : α →ᵇ 𝕜) (g : α →ᵇ β), of_normed_add_comm_group (λ x, (f x) • (g x))
(f.continuous.smul g.continuous) (‖f‖ * ‖g‖) (λ x, calc
‖f x • g x‖ ≤ ‖f x‖ * ‖g x‖ : norm_smul_le _ _
... ≤ ‖f‖ * ‖g‖ : mul_le_mul (f.norm_coe_le_norm _) (g.norm_coe_le_norm _) (norm_nonneg _)
(norm_nonneg _)) ⟩ | instance | bounded_continuous_function.has_smul' | 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",
"mul_le_mul",
"norm_smul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module' : module (α →ᵇ 𝕜) (α →ᵇ β) | module.of_core $
{ smul := (•),
smul_add := λ c f₁ f₂, ext $ λ x, smul_add _ _ _,
add_smul := λ c₁ c₂ f, ext $ λ x, add_smul _ _ _,
mul_smul := λ c₁ c₂ f, ext $ λ x, mul_smul _ _ _,
one_smul := λ f, ext $ λ x, one_smul 𝕜 (f x) } | instance | bounded_continuous_function.module' | 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_smul",
"module",
"module.of_core",
"one_smul",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_smul_le (f : α →ᵇ 𝕜) (g : α →ᵇ β) : ‖f • g‖ ≤ ‖f‖ * ‖g‖ | norm_of_normed_add_comm_group_le _ (mul_nonneg (norm_nonneg _) (norm_nonneg _)) _ | lemma | bounded_continuous_function.norm_smul_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"
] | [
"norm_smul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnreal.upper_bound {α : Type*} [topological_space α]
(f : α →ᵇ ℝ≥0) (x : α) : f x ≤ nndist f 0 | begin
have key : nndist (f x) ((0 : α →ᵇ ℝ≥0) x) ≤ nndist f 0,
{ exact @dist_coe_le_dist α ℝ≥0 _ _ f 0 x, },
simp only [coe_zero, pi.zero_apply] at key,
rwa nnreal.nndist_zero_eq_val' (f x) at key,
end | lemma | bounded_continuous_function.nnreal.upper_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"
] | [
"nnreal.nndist_zero_eq_val'",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_star (f : α →ᵇ β) : ⇑(star f) = star f | rfl | lemma | bounded_continuous_function.coe_star | 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 right-hand side of this equality can be parsed `star ∘ ⇑f` because of the
instance `pi.has_star`. Upon inspecting the goal, one sees `⊢ ⇑(star f) = star ⇑f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_apply (f : α →ᵇ β) (x : α) : star f x = star (f x) | rfl | lemma | bounded_continuous_function.star_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_sup (f g : α →ᵇ β) : ⇑(f ⊔ g) = f ⊔ g | rfl | lemma | bounded_continuous_function.coe_fn_sup | 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_abs (f : α →ᵇ β) : ⇑|f| = |f| | rfl | lemma | bounded_continuous_function.coe_fn_abs | 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 | |
nnreal_part (f : α →ᵇ ℝ) : α →ᵇ ℝ≥0 | bounded_continuous_function.comp _
(show lipschitz_with 1 real.to_nnreal, from lipschitz_with_pos) f | def | bounded_continuous_function.nnreal_part | 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.comp",
"lipschitz_with",
"lipschitz_with_pos",
"real.to_nnreal"
] | The nonnegative part of a bounded continuous `ℝ`-valued function as a bounded
continuous `ℝ≥0`-valued function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnreal_part_coe_fun_eq (f : α →ᵇ ℝ) : ⇑(f.nnreal_part) = real.to_nnreal ∘ ⇑f | rfl | lemma | bounded_continuous_function.nnreal_part_coe_fun_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"
] | [
"real.to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm (f : α →ᵇ ℝ) : α →ᵇ ℝ≥0 | bounded_continuous_function.comp _
(show lipschitz_with 1 (λ (x : ℝ), ‖x‖₊), from lipschitz_with_one_norm) f | def | bounded_continuous_function.nnnorm | 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.comp",
"lipschitz_with"
] | The absolute value of a bounded continuous `ℝ`-valued function as a bounded
continuous `ℝ≥0`-valued function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_coe_fun_eq (f : α →ᵇ ℝ) : ⇑(f.nnnorm) = has_nnnorm.nnnorm ∘ ⇑f | rfl | lemma | bounded_continuous_function.nnnorm_coe_fun_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 | |
self_eq_nnreal_part_sub_nnreal_part_neg (f : α →ᵇ ℝ) :
⇑f = coe ∘ f.nnreal_part - coe ∘ (-f).nnreal_part | by { funext x, dsimp, simp only [max_zero_sub_max_neg_zero_eq_self], } | lemma | bounded_continuous_function.self_eq_nnreal_part_sub_nnreal_part_neg | 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"
] | [] | Decompose a bounded continuous function to its positive and negative parts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_self_eq_nnreal_part_add_nnreal_part_neg (f : α →ᵇ ℝ) :
abs ∘ ⇑f = coe ∘ f.nnreal_part + coe ∘ (-f).nnreal_part | by { funext x, dsimp, simp only [max_zero_add_max_neg_zero_eq_abs_self], } | lemma | bounded_continuous_function.abs_self_eq_nnreal_part_add_nnreal_part_neg | 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"
] | [
"max_zero_add_max_neg_zero_eq_abs_self"
] | Express the absolute value of a bounded continuous function in terms of its
positive and negative parts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocompact_map (α : Type u) (β : Type v) [topological_space α] [topological_space β]
extends continuous_map α β : Type (max u v) | (cocompact_tendsto' : tendsto to_fun (cocompact α) (cocompact β)) | structure | cocompact_map | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"continuous_map",
"topological_space"
] | A *cocompact continuous map* is a continuous function between topological spaces which
tends to the cocompact filter along the cocompact filter. Functions for which preimages of compact
sets are compact always satisfy this property, and the converse holds for cocompact continuous maps
when the codomain is Hausdorff (se... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocompact_map_class (F : Type*) (α β : out_param $ Type*) [topological_space α]
[topological_space β] extends continuous_map_class F α β | (cocompact_tendsto (f : F) : tendsto f (cocompact α) (cocompact β)) | class | cocompact_map_class | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"continuous_map_class",
"topological_space"
] | `cocompact_map_class F α β` states that `F` is a type of cocompact continuous maps.
You should also extend this typeclass when you extend `cocompact_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_continuous_fun {f : cocompact_map α β} :
(f.to_continuous_map : α → β) = f | rfl | lemma | cocompact_map.coe_to_continuous_fun | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : cocompact_map α β} (h : ∀ x, f x = g x) : f = g | fun_like.ext _ _ h | lemma | cocompact_map.ext | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : cocompact_map α β) (f' : α → β) (h : f' = f) : cocompact_map α β | { to_fun := f',
continuous_to_fun := by {rw h, exact f.continuous_to_fun},
cocompact_tendsto' := by { simp_rw h, exact f.cocompact_tendsto' } } | def | cocompact_map.copy | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map"
] | Copy of a `cocompact_map` with a new `to_fun` equal to the old one. Useful
to fix definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : cocompact_map α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | cocompact_map.coe_copy | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : cocompact_map α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | cocompact_map.copy_eq | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : C(α, β)) (h : tendsto f (cocompact α) (cocompact β)) :
⇑(⟨f, h⟩ : cocompact_map α β) = f | rfl | lemma | cocompact_map.coe_mk | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : cocompact_map α α | ⟨continuous_map.id _, tendsto_id⟩ | def | cocompact_map.id | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map"
] | The identity as a cocompact continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(cocompact_map.id α) = id | rfl | lemma | cocompact_map.coe_id | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : cocompact_map β γ) (g : cocompact_map α β) : cocompact_map α γ | ⟨f.to_continuous_map.comp g, (cocompact_tendsto f).comp (cocompact_tendsto g)⟩ | def | cocompact_map.comp | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map"
] | The composition of cocompact continuous maps, as a cocompact continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : cocompact_map β γ) (g : cocompact_map α β) :
⇑(comp f g) = f ∘ g | rfl | lemma | cocompact_map.coe_comp | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : cocompact_map β γ) (g : cocompact_map α β) (a : α) :
comp f g a = f (g a) | rfl | lemma | cocompact_map.comp_apply | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : cocompact_map γ δ) (g : cocompact_map β γ)
(h : cocompact_map α β) : (f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | cocompact_map.comp_assoc | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : cocompact_map α β) : (cocompact_map.id _).comp f = f | ext $ λ _, rfl | lemma | cocompact_map.id_comp | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map",
"cocompact_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : cocompact_map α β) : f.comp (cocompact_map.id _) = f | ext $ λ _, rfl | lemma | cocompact_map.comp_id | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map",
"cocompact_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_of_forall_preimage {f : α → β} (h : ∀ s, is_compact s → is_compact (f ⁻¹' s)) :
tendsto f (cocompact α) (cocompact β) | λ s hs, match mem_cocompact.mp hs with ⟨t, ht, hts⟩ :=
mem_map.mpr (mem_cocompact.mpr ⟨f ⁻¹' t, h t ht, by simpa using preimage_mono hts⟩) end | lemma | cocompact_map.tendsto_of_forall_preimage | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_preimage [t2_space β] (f : cocompact_map α β) ⦃s : set β⦄ (hs : is_compact s) :
is_compact (f ⁻¹' s) | begin
obtain ⟨t, ht, hts⟩ := mem_cocompact'.mp (by simpa only [preimage_image_preimage, preimage_compl]
using mem_map.mp (cocompact_tendsto f $ mem_cocompact.mpr ⟨s, hs, compl_subset_compl.mpr
(image_preimage_subset f _)⟩)),
exact is_compact_of_is_closed_subset ht (hs.is_closed.preimage $ map_continuous f)
... | lemma | cocompact_map.is_compact_preimage | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map",
"is_compact",
"is_compact_of_is_closed_subset",
"t2_space"
] | If the codomain is Hausdorff, preimages of compact sets are compact under a cocompact
continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homeomorph.to_cocompact_map
{α β : Type*} [topological_space α] [topological_space β] (f : α ≃ₜ β) : cocompact_map α β | { to_fun := f,
continuous_to_fun := f.continuous,
cocompact_tendsto' :=
begin
refine cocompact_map.tendsto_of_forall_preimage (λ K hK, _),
erw K.preimage_equiv_eq_image_symm,
exact hK.image f.symm.continuous,
end } | def | homeomorph.to_cocompact_map | topology.continuous_function | src/topology/continuous_function/cocompact_map.lean | [
"topology.continuous_function.basic"
] | [
"cocompact_map",
"cocompact_map.tendsto_of_forall_preimage",
"topological_space"
] | A homemomorphism is a cocompact map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_bounded_of_compact : C(α, β) ≃ (α →ᵇ β) | ⟨mk_of_compact, bounded_continuous_function.to_continuous_map,
λ f, by { ext, refl, }, λ f, by { ext, refl, }⟩ | def | continuous_map.equiv_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, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing_equiv_bounded_of_compact :
uniform_inducing (equiv_bounded_of_compact α β) | uniform_inducing.mk'
begin
simp only [has_basis_compact_convergence_uniformity.mem_iff, uniformity_basis_dist_le.mem_iff],
exact λ s, ⟨λ ⟨⟨a, b⟩, ⟨ha, ⟨ε, hε, hb⟩⟩, hs⟩, ⟨{p | ∀ x, (p.1 x, p.2 x) ∈ b},
⟨ε, hε, λ _ h x, hb (by exact (dist_le hε.le).mp h x)⟩, λ f g h, hs (by exact λ x hx, h x)⟩,
λ ⟨t, ⟨ε, hε,... | lemma | continuous_map.uniform_inducing_equiv_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"
] | [
"uniform_inducing",
"uniform_inducing.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_equiv_bounded_of_compact :
uniform_embedding (equiv_bounded_of_compact α β) | { inj := (equiv_bounded_of_compact α β).injective,
.. uniform_inducing_equiv_bounded_of_compact α β } | lemma | continuous_map.uniform_embedding_equiv_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"
] | [
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_equiv_bounded_of_compact [add_monoid β] [has_lipschitz_add β] :
C(α, β) ≃+ (α →ᵇ β) | ({ .. to_continuous_map_add_hom α β,
.. (equiv_bounded_of_compact α β).symm, } : (α →ᵇ β) ≃+ C(α, β)).symm | def | continuous_map.add_equiv_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"
] | [
"add_monoid",
"has_lipschitz_add"
] | When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_equiv_bounded_of_compact :
C(α, β) ≃ᵢ (α →ᵇ β) | { isometry_to_fun := λ x y, rfl,
to_equiv := equiv_bounded_of_compact α β } | def | continuous_map.isometry_equiv_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 `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.bounded_continuous_function.dist_mk_of_compact (f g : C(α, β)) :
dist (mk_of_compact f) (mk_of_compact g) = dist f g | rfl | lemma | bounded_continuous_function.dist_mk_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.bounded_continuous_function.dist_to_continuous_map (f g : α →ᵇ β) :
dist (f.to_continuous_map) (g.to_continuous_map) = dist f g | rfl | lemma | bounded_continuous_function.dist_to_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g | by simp only [← dist_mk_of_compact, dist_coe_le_dist, ← mk_of_compact_apply] | lemma | continuous_map.dist_apply_le_dist | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | The pointwise distance is controlled by the distance between functions, by definition. | 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 | by simp only [← dist_mk_of_compact, dist_le C0, mk_of_compact_apply] | lemma | continuous_map.dist_le | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | 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 | by simp only [← dist_mk_of_compact, dist_le_iff_of_nonempty, mk_of_compact_apply] | lemma | continuous_map.dist_le_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 | |
dist_lt_iff_of_nonempty [nonempty α] :
dist f g < C ↔ ∀x:α, dist (f x) (g x) < C | by simp only [← dist_mk_of_compact, dist_lt_iff_of_nonempty_compact, mk_of_compact_apply] | lemma | continuous_map.dist_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 | |
dist_lt_of_nonempty [nonempty α] (w : ∀x:α, dist (f x) (g x) < C) : dist f g < C | (dist_lt_iff_of_nonempty).2 w | lemma | continuous_map.dist_lt_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 | |
dist_lt_iff (C0 : (0 : ℝ) < C) :
dist f g < C ↔ ∀x:α, dist (f x) (g x) < C | by simp only [← dist_mk_of_compact, dist_lt_iff_of_compact C0, mk_of_compact_apply] | lemma | continuous_map.dist_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 | |
continuous_eval : continuous (λ p : C(α, β) × α, p.1 p.2) | continuous_eval.comp ((isometry_equiv_bounded_of_compact α β).continuous.prod_map continuous_id) | lemma | continuous_map.continuous_eval | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"continuous",
"continuous.prod_map",
"continuous_id"
] | See also `continuous_map.continuous_eval'` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_eval_const (x : α) : continuous (λ f : C(α, β), f x) | continuous_eval.comp (continuous_id.prod_mk continuous_const) | lemma | continuous_map.continuous_eval_const | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"continuous",
"continuous_const"
] | See also `continuous_map.continuous_eval_const` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_coe : @continuous (C(α, β)) (α → β) _ _ coe_fn | continuous_pi continuous_eval_const | lemma | continuous_map.continuous_coe | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [
"continuous",
"continuous_pi"
] | See also `continuous_map.continuous_coe'` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.bounded_continuous_function.norm_mk_of_compact (f : C(α, E)) :
‖mk_of_compact f‖ = ‖f‖ | rfl | lemma | bounded_continuous_function.norm_mk_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.bounded_continuous_function.norm_to_continuous_map_eq (f : α →ᵇ E) :
‖f.to_continuous_map‖ = ‖f‖ | rfl | lemma | bounded_continuous_function.norm_to_continuous_map_eq | 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_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ | (mk_of_compact f).norm_coe_le_norm x | lemma | continuous_map.norm_coe_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 | |
dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ | (mk_of_compact f).dist_le_two_norm x y | lemma | continuous_map.dist_le_two_norm | topology.continuous_function | src/topology/continuous_function/compact.lean | [
"topology.continuous_function.bounded",
"topology.uniform_space.compact",
"topology.compact_open",
"topology.sets.compacts"
] | [] | 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 {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀x:α, ‖f x‖ ≤ C | @bounded_continuous_function.norm_le _ _ _ _
(mk_of_compact f) _ C0 | lemma | continuous_map.norm_le | 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"
] | The norm of a function is controlled by the supremum of the pointwise norms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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