statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
coe_fn_of_bijective (f : E →L[𝕜] F) (hinj : ker f = ⊥)
(hsurj : linear_map.range f = ⊤) : ⇑(of_bijective f hinj hsurj) = f | rfl | lemma | continuous_linear_equiv.coe_fn_of_bijective | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"linear_map.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_bijective (f : E →L[𝕜] F) (hinj : ker f = ⊥) (hsurj : linear_map.range f = ⊤) :
↑(of_bijective f hinj hsurj) = f | by { ext, refl } | lemma | continuous_linear_equiv.coe_of_bijective | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"linear_map.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bijective_symm_apply_apply (f : E →L[𝕜] F) (hinj : ker f = ⊥)
(hsurj : linear_map.range f = ⊤) (x : E) :
(of_bijective f hinj hsurj).symm (f x) = x | (of_bijective f hinj hsurj).symm_apply_apply x | lemma | continuous_linear_equiv.of_bijective_symm_apply_apply | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"linear_map.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bijective_apply_symm_apply (f : E →L[𝕜] F) (hinj : ker f = ⊥)
(hsurj : linear_map.range f = ⊤) (y : F) :
f ((of_bijective f hinj hsurj).symm y) = y | (of_bijective f hinj hsurj).apply_symm_apply y | lemma | continuous_linear_equiv.of_bijective_apply_symm_apply | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"linear_map.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_subtypeL_equiv_of_is_compl
(f : E →L[𝕜] F) {G : submodule 𝕜 F}
(h : is_compl (linear_map.range f) G) [complete_space G] (hker : ker f = ⊥) : (E × G) ≃L[𝕜] F | continuous_linear_equiv.of_bijective (f.coprod G.subtypeL)
(begin
rw ker_coprod_of_disjoint_range,
{ rw [hker, submodule.ker_subtypeL, submodule.prod_bot] },
{ rw submodule.range_subtypeL,
exact h.disjoint }
end)
(by simp only [range_coprod, h.sup_eq_top, submodule.range_subtypeL]) | def | continuous_linear_map.coprod_subtypeL_equiv_of_is_compl | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"complete_space",
"continuous_linear_equiv.of_bijective",
"is_compl",
"linear_map.range",
"submodule",
"submodule.ker_subtypeL",
"submodule.prod_bot",
"submodule.range_subtypeL"
] | Intermediate definition used to show
`continuous_linear_map.closed_complemented_range_of_is_compl_of_ker_eq_bot`.
This is `f.coprod G.subtypeL` as an `continuous_linear_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
range_eq_map_coprod_subtypeL_equiv_of_is_compl
(f : E →L[𝕜] F) {G : submodule 𝕜 F}
(h : is_compl (linear_map.range f) G) [complete_space G] (hker : ker f = ⊥) :
linear_map.range f = ((⊤ : submodule 𝕜 E).prod (⊥ : submodule 𝕜 G)).map
(f.coprod_subtypeL_equiv_of_is_compl h hker : E × G →ₗ[𝕜] F) | begin
rw [coprod_subtypeL_equiv_of_is_compl, _root_.coe_coe, continuous_linear_equiv.coe_of_bijective,
coe_coprod, linear_map.coprod_map_prod, submodule.map_bot, sup_bot_eq, submodule.map_top],
refl
end | lemma | continuous_linear_map.range_eq_map_coprod_subtypeL_equiv_of_is_compl | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"complete_space",
"continuous_linear_equiv.coe_of_bijective",
"is_compl",
"linear_map.coprod_map_prod",
"linear_map.range",
"submodule",
"submodule.map_bot",
"submodule.map_top",
"sup_bot_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_complemented_range_of_is_compl_of_ker_eq_bot (f : E →L[𝕜] F) (G : submodule 𝕜 F)
(h : is_compl (linear_map.range f) G) (hG : is_closed (G : set F)) (hker : ker f = ⊥) :
is_closed (linear_map.range f : set F) | begin
haveI : complete_space G := hG.complete_space_coe,
let g := coprod_subtypeL_equiv_of_is_compl f h hker,
rw congr_arg coe (range_eq_map_coprod_subtypeL_equiv_of_is_compl f h hker ),
apply g.to_homeomorph.is_closed_image.2,
exact is_closed_univ.prod is_closed_singleton,
end | lemma | continuous_linear_map.closed_complemented_range_of_is_compl_of_ker_eq_bot | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"complete_space",
"is_closed",
"is_closed_singleton",
"is_compl",
"linear_map.range",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.continuous_of_is_closed_graph (hg : is_closed (g.graph : set $ E × F)) :
continuous g | begin
letI : complete_space g.graph := complete_space_coe_iff_is_complete.mpr hg.is_complete,
let φ₀ : E →ₗ[𝕜] E × F := linear_map.id.prod g,
have : function.left_inverse prod.fst φ₀ := λ x, rfl,
let φ : E ≃ₗ[𝕜] g.graph :=
(linear_equiv.of_left_inverse this).trans
(linear_equiv.of_eq _ _ g.graph_eq_ra... | theorem | linear_map.continuous_of_is_closed_graph | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"complete_space",
"continuous",
"is_closed",
"linear_equiv.of_eq",
"linear_equiv.of_left_inverse"
] | The **closed graph theorem** : a linear map between two Banach spaces whose graph is closed
is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.continuous_of_seq_closed_graph
(hg : ∀ (u : ℕ → E) x y, tendsto u at_top (𝓝 x) → tendsto (g ∘ u) at_top (𝓝 y) → y = g x) :
continuous g | begin
refine g.continuous_of_is_closed_graph (is_seq_closed.is_closed _),
rintros φ ⟨x, y⟩ hφg hφ,
refine hg (prod.fst ∘ φ) x y ((continuous_fst.tendsto _).comp hφ) _,
have : g ∘ prod.fst ∘ φ = prod.snd ∘ φ,
{ ext n,
exact (hφg n).symm },
rw this,
exact (continuous_snd.tendsto _).comp hφ
end | theorem | linear_map.continuous_of_seq_closed_graph | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"continuous",
"is_seq_closed.is_closed"
] | A useful form of the **closed graph theorem** : let `f` be a linear map between two Banach
spaces. To show that `f` is continuous, it suffices to show that for any convergent sequence
`uₙ ⟶ x`, if `f(uₙ) ⟶ y` then `y = f(x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_closed_graph (hg : is_closed (g.graph : set $ E × F)) :
E →L[𝕜] F | { to_linear_map := g,
cont := g.continuous_of_is_closed_graph hg } | def | continuous_linear_map.of_is_closed_graph | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"cont",
"is_closed"
] | Upgrade a `linear_map` to a `continuous_linear_map` using the **closed graph theorem**. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_of_is_closed_graph (hg : is_closed (g.graph : set $ E × F)) :
⇑(continuous_linear_map.of_is_closed_graph hg) = g | rfl | lemma | continuous_linear_map.coe_fn_of_is_closed_graph | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"continuous_linear_map.of_is_closed_graph",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_is_closed_graph (hg : is_closed (g.graph : set $ E × F)) :
↑(continuous_linear_map.of_is_closed_graph hg) = g | by { ext, refl } | lemma | continuous_linear_map.coe_of_is_closed_graph | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"continuous_linear_map.of_is_closed_graph",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_seq_closed_graph
(hg : ∀ (u : ℕ → E) x y, tendsto u at_top (𝓝 x) → tendsto (g ∘ u) at_top (𝓝 y) → y = g x) :
E →L[𝕜] F | { to_linear_map := g,
cont := g.continuous_of_seq_closed_graph hg } | def | continuous_linear_map.of_seq_closed_graph | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"cont"
] | Upgrade a `linear_map` to a `continuous_linear_map` using a variation on the
**closed graph theorem**. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_of_seq_closed_graph
(hg : ∀ (u : ℕ → E) x y, tendsto u at_top (𝓝 x) → tendsto (g ∘ u) at_top (𝓝 y) → y = g x) :
⇑(continuous_linear_map.of_seq_closed_graph hg) = g | rfl | lemma | continuous_linear_map.coe_fn_of_seq_closed_graph | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"continuous_linear_map.of_seq_closed_graph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_seq_closed_graph
(hg : ∀ (u : ℕ → E) x y, tendsto u at_top (𝓝 x) → tendsto (g ∘ u) at_top (𝓝 y) → y = g x) :
↑(continuous_linear_map.of_seq_closed_graph hg) = g | by { ext, refl } | lemma | continuous_linear_map.coe_of_seq_closed_graph | analysis.normed_space | src/analysis/normed_space/banach.lean | [
"topology.metric_space.baire",
"analysis.normed_space.operator_norm",
"analysis.normed_space.affine_isometry"
] | [
"continuous_linear_map.of_seq_closed_graph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
banach_steinhaus {ι : Type*} [complete_space E] {g : ι → E →SL[σ₁₂] F}
(h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) :
∃ C', ∀ i, ‖g i‖ ≤ C' | begin
/- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` -/
let e : ℕ → set E := λ n, (⋂ i : ι, { x : E | ‖g i x‖ ≤ n }),
/- each of these sets is closed -/
have hc : ∀ n : ℕ, is_closed (e n), from λ i, is_closed_Inter (λ i,
is_closed_le (continuous.norm (g i).cont) conti... | theorem | banach_steinhaus | analysis.normed_space | src/analysis/normed_space/banach_steinhaus.lean | [
"analysis.normed_space.operator_norm",
"topology.metric_space.baire",
"topology.algebra.module.basic"
] | [
"complete_space",
"cont",
"continuous_const",
"continuous_linear_map.map_add",
"continuous_linear_map.op_norm_le_of_shell",
"div_nonneg",
"div_pos",
"exists_nat_ge",
"interior_Inter_subset",
"interior_subset",
"is_closed",
"is_closed_Inter",
"is_closed_le",
"is_open_interior",
"le_mul_of... | This is the standard Banach-Steinhaus theorem, or Uniform Boundedness Principle.
If a family of continuous linear maps from a Banach space into a normed space is pointwise
bounded, then the norms of these linear maps are uniformly bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
banach_steinhaus_supr_nnnorm {ι : Type*} [complete_space E] {g : ι → E →SL[σ₁₂] F}
(h : ∀ x, (⨆ i, ↑‖g i x‖₊) < ∞) :
(⨆ i, ↑‖g i‖₊) < ∞ | begin
have h' : ∀ x : E, ∃ C : ℝ, ∀ i : ι, ‖g i x‖ ≤ C,
{ intro x,
rcases lt_iff_exists_coe.mp (h x) with ⟨p, hp₁, _⟩,
refine ⟨p, (λ i, _)⟩,
exact_mod_cast
calc (‖g i x‖₊ : ℝ≥0∞) ≤ ⨆ j, ‖g j x‖₊ : le_supr _ i
... = p : hp₁ },
cases banach_steinhaus h' with C' h... | theorem | banach_steinhaus_supr_nnnorm | analysis.normed_space | src/analysis/normed_space/banach_steinhaus.lean | [
"analysis.normed_space.operator_norm",
"topology.metric_space.baire",
"topology.algebra.module.basic"
] | [
"banach_steinhaus",
"complete_space",
"le_supr",
"real.to_nnreal_le_to_nnreal",
"supr_le"
] | This version of Banach-Steinhaus is stated in terms of suprema of `↑‖⬝‖₊ : ℝ≥0∞`
for convenience. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map_of_tendsto [complete_space E] [t2_space F]
(g : ℕ → E →SL[σ₁₂] F) {f : E → F} (h : tendsto (λ n x, g n x) at_top (𝓝 f)) :
E →SL[σ₁₂] F | { to_fun := f,
map_add' := (linear_map_of_tendsto _ _ h).map_add',
map_smul' := (linear_map_of_tendsto _ _ h).map_smul',
cont :=
begin
/- show that the maps are pointwise bounded and apply `banach_steinhaus`-/
have h_point_bdd : ∀ x : E, ∃ C : ℝ, ∀ n : ℕ, ‖g n x‖ ≤ C,
{ intro x,
rcas... | def | continuous_linear_map_of_tendsto | analysis.normed_space | src/analysis/normed_space/banach_steinhaus.lean | [
"analysis.normed_space.operator_norm",
"topology.metric_space.baire",
"topology.algebra.module.basic"
] | [
"banach_steinhaus",
"cauchy_seq",
"cauchy_seq_bdd",
"complete_space",
"cont",
"linear_map_of_tendsto",
"t2_space"
] | Given a *sequence* of continuous linear maps which converges pointwise and for which the
domain is complete, the Banach-Steinhaus theorem is used to guarantee that the limit map
is a *continuous* linear map as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_space (α : Type*) (β : Type*) [normed_field α] [seminormed_add_comm_group β]
extends module α β | (norm_smul_le : ∀ (a:α) (b:β), ‖a • b‖ ≤ ‖a‖ * ‖b‖) | class | normed_space | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"module",
"norm_smul_le",
"normed_field",
"seminormed_add_comm_group"
] | A normed space over a normed field is a vector space endowed with a norm which satisfies the
equality `‖c • x‖ = ‖c‖ ‖x‖`. We require only `‖c • x‖ ≤ ‖c‖ ‖x‖` in the definition, then prove
`‖c • x‖ = ‖c‖ ‖x‖` in `norm_smul`.
Note that since this requires `seminormed_add_comm_group` and not `normed_add_comm_group`, thi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_space.has_bounded_smul [normed_space α β] : has_bounded_smul α β | has_bounded_smul.of_norm_smul_le normed_space.norm_smul_le | instance | normed_space.has_bounded_smul | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"has_bounded_smul",
"has_bounded_smul.of_norm_smul_le",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_field.to_normed_space : normed_space α α | { norm_smul_le := λ a b, norm_mul_le a b } | instance | normed_field.to_normed_space | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"norm_mul_le",
"norm_smul_le",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_field.to_has_bounded_smul : has_bounded_smul α α | normed_space.has_bounded_smul | instance | normed_field.to_has_bounded_smul | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"has_bounded_smul",
"normed_space.has_bounded_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_zsmul (α) [normed_field α] [normed_space α β] (n : ℤ) (x : β) :
‖n • x‖ = ‖(n : α)‖ * ‖x‖ | by rw [← norm_smul, ← int.smul_one_eq_coe, smul_assoc, one_smul] | lemma | norm_zsmul | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"int.smul_one_eq_coe",
"norm_smul",
"normed_field",
"normed_space",
"one_smul",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_norm (z : β) : |‖z‖| = ‖z‖ | abs_of_nonneg $ norm_nonneg z | lemma | abs_norm | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"abs_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_norm_smul_mem_closed_unit_ball [normed_space ℝ β] (x : β) :
‖x‖⁻¹ • x ∈ closed_ball (0 : β) 1 | by simp only [mem_closed_ball_zero_iff, norm_smul, norm_inv, norm_norm, ← div_eq_inv_mul,
div_self_le_one] | lemma | inv_norm_smul_mem_closed_unit_ball | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"div_eq_inv_mul",
"div_self_le_one",
"norm_inv",
"norm_norm",
"norm_smul",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_smul_of_nonneg [normed_space ℝ β] {t : ℝ} (ht : 0 ≤ t) (x : β) :
‖t • x‖ = t * ‖x‖ | by rw [norm_smul, real.norm_eq_abs, abs_of_nonneg ht] | lemma | norm_smul_of_nonneg | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"abs_of_nonneg",
"norm_smul",
"normed_space",
"real.norm_eq_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_norm_smul_sub_lt (c : α) (x : E) {ε : ℝ} (h : 0 < ε) :
∀ᶠ y in 𝓝 x, ‖c • (y - x)‖ < ε | have tendsto (λ y, ‖c • (y - x)‖) (𝓝 x) (𝓝 0),
from ((continuous_id.sub continuous_const).const_smul _).norm.tendsto' _ _ (by simp),
this.eventually (gt_mem_nhds h) | theorem | eventually_nhds_norm_smul_sub_lt | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"continuous_const",
"gt_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.zero_smul_is_bounded_under_le {f : ι → α} {g : ι → E} {l : filter ι}
(hf : tendsto f l (𝓝 0)) (hg : is_bounded_under (≤) l (norm ∘ g)) :
tendsto (λ x, f x • g x) l (𝓝 0) | hf.op_zero_is_bounded_under_le hg (•) norm_smul_le | lemma | filter.tendsto.zero_smul_is_bounded_under_le | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"filter",
"norm_smul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.is_bounded_under.smul_tendsto_zero {f : ι → α} {g : ι → E} {l : filter ι}
(hf : is_bounded_under (≤) l (norm ∘ f)) (hg : tendsto g l (𝓝 0)) :
tendsto (λ x, f x • g x) l (𝓝 0) | hg.op_zero_is_bounded_under_le hf (flip (•)) (λ x y, (norm_smul_le y x).trans_eq (mul_comm _ _)) | lemma | filter.is_bounded_under.smul_tendsto_zero | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"filter",
"mul_comm",
"norm_smul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_ball [normed_space ℝ E] (x : E) {r : ℝ} (hr : r ≠ 0) :
closure (ball x r) = closed_ball x r | begin
refine subset.antisymm closure_ball_subset_closed_ball (λ y hy, _),
have : continuous_within_at (λ c : ℝ, c • (y - x) + x) (Ico 0 1) 1 :=
((continuous_id.smul continuous_const).add continuous_const).continuous_within_at,
convert this.mem_closure _ _,
{ rw [one_smul, sub_add_cancel] },
{ simp [closur... | theorem | closure_ball | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"abs_of_nonneg",
"closure",
"closure_Ico",
"continuous_const",
"continuous_within_at",
"mul_comm",
"mul_lt_mul'",
"mul_one",
"norm_smul",
"normed_space",
"one_smul",
"real.norm_eq_abs",
"zero_le_one",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_ball [normed_space ℝ E] (x : E) {r : ℝ} (hr : r ≠ 0) :
frontier (ball x r) = sphere x r | begin
rw [frontier, closure_ball x hr, is_open_ball.interior_eq],
ext x, exact (@eq_iff_le_not_lt ℝ _ _ _).symm
end | theorem | frontier_ball | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"closure_ball",
"eq_iff_le_not_lt",
"frontier",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_closed_ball [normed_space ℝ E] (x : E) {r : ℝ} (hr : r ≠ 0) :
interior (closed_ball x r) = ball x r | begin
cases hr.lt_or_lt with hr hr,
{ rw [closed_ball_eq_empty.2 hr, ball_eq_empty.2 hr.le, interior_empty] },
refine subset.antisymm _ ball_subset_interior_closed_ball,
intros y hy,
rcases (mem_closed_ball.1 $ interior_subset hy).lt_or_eq with hr|rfl, { exact hr },
set f : ℝ → E := λ c : ℝ, c • (y - x) + x... | theorem | interior_closed_ball | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"abs_le",
"continuous",
"continuous_const",
"interior",
"interior_empty",
"interior_mono",
"interior_subset",
"mul_le_mul_right",
"norm_smul",
"normed_space",
"preimage_interior_subset_interior_preimage",
"real.norm_eq_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_closed_ball [normed_space ℝ E] (x : E) {r : ℝ} (hr : r ≠ 0) :
frontier (closed_ball x r) = sphere x r | by rw [frontier, closure_closed_ball, interior_closed_ball x hr,
closed_ball_diff_ball] | theorem | frontier_closed_ball | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"frontier",
"interior_closed_ball",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_sphere [normed_space ℝ E] (x : E) {r : ℝ} (hr : r ≠ 0) :
interior (sphere x r) = ∅ | by rw [←frontier_closed_ball x hr, interior_frontier is_closed_ball] | theorem | interior_sphere | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"interior",
"interior_frontier",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_sphere [normed_space ℝ E] (x : E) {r : ℝ} (hr : r ≠ 0) :
frontier (sphere x r) = sphere x r | by rw [is_closed_sphere.frontier_eq, interior_sphere x hr, diff_empty] | theorem | frontier_sphere | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"frontier",
"interior_sphere",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph_unit_ball [normed_space ℝ E] :
E ≃ₜ ball (0 : E) 1 | { to_fun := λ x, ⟨(1 + ‖x‖^2).sqrt⁻¹ • x, begin
have : 0 < 1 + ‖x‖ ^ 2, by positivity,
rw [mem_ball_zero_iff, norm_smul, real.norm_eq_abs, abs_inv, ← div_eq_inv_mul,
div_lt_one (abs_pos.mpr $ real.sqrt_ne_zero'.mpr this), ← abs_norm x, ← sq_lt_sq,
abs_norm, real.sq_sqrt this.le],
exact lt_one_ad... | def | homeomorph_unit_ball | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"abs_inv",
"abs_norm",
"continuity",
"continuous",
"continuous.inv₀",
"continuous_id",
"div_eq_inv_mul",
"div_lt_one",
"inv_fun",
"lt_one_add",
"norm_smul",
"normed_space",
"real.norm_eq_abs",
"real.sq_sqrt",
"real.sqrt_div",
"real.sqrt_ne_zero'",
"smul_smul",
"sq_lt_sq"
] | A (semi) normed real vector space is homeomorphic to the unit ball in the same space.
This homeomorphism sends `x : E` to `(1 + ‖x‖²)^(- ½) • x`.
In many cases the actual implementation is not important, so we don't mark the projection lemmas
`homeomorph_unit_ball_apply_coe` and `homeomorph_unit_ball_symm_apply` as `@... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_homeomorph_unit_ball_apply_zero [normed_space ℝ E] :
(homeomorph_unit_ball (0 : E) : E) = 0 | by simp [homeomorph_unit_ball] | lemma | coe_homeomorph_unit_ball_apply_zero | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"homeomorph_unit_ball",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.normed_space : normed_space α (E × F) | { norm_smul_le := λ s x, by simp [prod.norm_def, norm_smul_le, mul_max_of_nonneg],
..prod.normed_add_comm_group,
..prod.module } | instance | prod.normed_space | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"mul_max_of_nonneg",
"norm_smul_le",
"normed_space",
"prod.norm_def"
] | The product of two normed spaces is a normed space, with the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.normed_space {E : ι → Type*} [fintype ι] [∀i, seminormed_add_comm_group (E i)]
[∀i, normed_space α (E i)] : normed_space α (Πi, E i) | { norm_smul_le := λ a f, begin
simp_rw [←coe_nnnorm, ←nnreal.coe_mul, nnreal.coe_le_coe, pi.nnnorm_def, nnreal.mul_finset_sup],
exact finset.sup_mono_fun (λ _ _, norm_smul_le _ _),
end } | instance | pi.normed_space | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"finset.sup_mono_fun",
"fintype",
"nnreal.coe_le_coe",
"nnreal.mul_finset_sup",
"norm_smul_le",
"normed_space",
"seminormed_add_comm_group"
] | The product of finitely many normed spaces is a normed space, with the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_opposite.normed_space : normed_space α Eᵐᵒᵖ | { norm_smul_le := λ s x, (norm_smul_le s x.unop : _),
..mul_opposite.normed_add_comm_group,
..mul_opposite.module _ } | instance | mul_opposite.normed_space | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"norm_smul_le",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.normed_space {𝕜 R : Type*} [has_smul 𝕜 R] [normed_field 𝕜] [ring R]
{E : Type*} [seminormed_add_comm_group E] [normed_space 𝕜 E] [module R E]
[is_scalar_tower 𝕜 R E] (s : submodule R E) :
normed_space 𝕜 s | { norm_smul_le := λc x, (norm_smul_le c (x : E) : _) } | instance | submodule.normed_space | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"has_smul",
"is_scalar_tower",
"module",
"norm_smul_le",
"normed_field",
"normed_space",
"ring",
"seminormed_add_comm_group",
"submodule"
] | A subspace of a normed space is also a normed space, with the restriction of the norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rescale_to_shell_semi_normed_zpow {c : α} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E}
(hx : ‖x‖ ≠ 0) :
∃ n : ℤ, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ (ε / ‖c‖ ≤ ‖c ^ n • x‖) ∧ (‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖) | begin
have xεpos : 0 < ‖x‖/ε := div_pos ((ne.symm hx).le_iff_lt.1 (norm_nonneg x)) εpos,
rcases exists_mem_Ico_zpow xεpos hc with ⟨n, hn⟩,
have cpos : 0 < ‖c‖ := lt_trans (zero_lt_one : (0 :ℝ) < 1) hc,
have cnpos : 0 < ‖c^(n+1)‖ := by { rw norm_zpow, exact lt_trans xεpos hn.2 },
refine ⟨-(n+1), _, _, _, _⟩,
... | lemma | rescale_to_shell_semi_normed_zpow | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"div_eq_inv_mul",
"div_le_iff",
"div_lt_iff",
"div_pos",
"exists_mem_Ico_zpow",
"inv_inv",
"le_div_iff",
"mul_assoc",
"mul_comm",
"mul_inv_cancel",
"mul_inv_rev",
"mul_le_mul_of_nonneg_right",
"mul_right_comm",
"norm_inv",
"norm_smul",
"norm_zpow",
"one_mul",
"zero_lt_one",
"zpow... | If there is a scalar `c` with `‖c‖>1`, then any element with nonzero norm can be
moved by scalar multiplication to any shell of width `‖c‖`. Also recap information on the norm of
the rescaling element that shows up in applications. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rescale_to_shell_semi_normed {c : α} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E}
(hx : ‖x‖ ≠ 0) : ∃d:α, d ≠ 0 ∧ ‖d • x‖ < ε ∧ (ε/‖c‖ ≤ ‖d • x‖) ∧ (‖d‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖) | let ⟨n, hn⟩ := rescale_to_shell_semi_normed_zpow hc εpos hx in ⟨_, hn⟩ | lemma | rescale_to_shell_semi_normed | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"rescale_to_shell_semi_normed_zpow"
] | If there is a scalar `c` with `‖c‖>1`, then any element with nonzero norm can be
moved by scalar multiplication to any shell of width `‖c‖`. Also recap information on the norm of
the rescaling element that shows up in applications. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_space.induced {F : Type*} (α β γ : Type*) [normed_field α] [add_comm_group β]
[module α β] [seminormed_add_comm_group γ] [normed_space α γ] [linear_map_class F α β γ]
(f : F) : @normed_space α β _ (seminormed_add_comm_group.induced β γ f) | { norm_smul_le := λ a b, by {unfold norm, exact (map_smul f a b).symm ▸ norm_smul_le a (f b) } } | def | normed_space.induced | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"add_comm_group",
"linear_map_class",
"module",
"norm_smul_le",
"normed_field",
"normed_space",
"seminormed_add_comm_group"
] | A linear map from a `module` to a `normed_space` induces a `normed_space` structure on the
domain, using the `seminormed_add_comm_group.induced` norm.
See note [reducible non-instances] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_space.to_module' : module α F | normed_space.to_module | instance | normed_space.to_module' | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"module"
] | While this may appear identical to `normed_space.to_module`, it contains an implicit argument
involving `normed_add_comm_group.to_seminormed_add_comm_group` that typeclass inference has trouble
inferring.
Specifically, the following instance cannot be found without this `normed_space.to_module'`:
```lean
example
(𝕜... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_norm_eq {c : ℝ} (hc : 0 ≤ c) : ∃ x : E, ‖x‖ = c | begin
rcases exists_ne (0 : E) with ⟨x, hx⟩,
rw ← norm_ne_zero_iff at hx,
use c • ‖x‖⁻¹ • x,
simp [norm_smul, real.norm_of_nonneg hc, hx]
end | lemma | exists_norm_eq | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"exists_ne",
"norm_smul",
"real.norm_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_norm : range (norm : E → ℝ) = Ici 0 | subset.antisymm (range_subset_iff.2 norm_nonneg) (λ _, exists_norm_eq E) | lemma | range_norm | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"exists_norm_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_surjective : surjective (nnnorm : E → ℝ≥0) | λ c, (exists_norm_eq E c.coe_nonneg).imp $ λ x h, nnreal.eq h | lemma | nnnorm_surjective | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"exists_norm_eq",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_nnnorm : range (nnnorm : E → ℝ≥0) = univ | (nnnorm_surjective E).range_eq | lemma | range_nnnorm | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"nnnorm_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.punctured_nhds_module_ne_bot
{E : Type*} [add_comm_group E] [topological_space E] [has_continuous_add E] [nontrivial E]
[module ℝ E] [has_continuous_smul ℝ E] (x : E) :
ne_bot (𝓝[≠] x) | module.punctured_nhds_ne_bot ℝ E x | instance | real.punctured_nhds_module_ne_bot | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"add_comm_group",
"has_continuous_add",
"has_continuous_smul",
"module",
"module.punctured_nhds_ne_bot",
"nontrivial",
"topological_space"
] | If `E` is a nontrivial topological module over `ℝ`, then `E` has no isolated points.
This is a particular case of `module.punctured_nhds_ne_bot`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interior_closed_ball' [normed_space ℝ E] [nontrivial E] (x : E) (r : ℝ) :
interior (closed_ball x r) = ball x r | begin
rcases eq_or_ne r 0 with rfl|hr,
{ rw [closed_ball_zero, ball_zero, interior_singleton] },
{ exact interior_closed_ball x hr }
end | theorem | interior_closed_ball' | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"eq_or_ne",
"interior",
"interior_closed_ball",
"interior_singleton",
"nontrivial",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_closed_ball' [normed_space ℝ E] [nontrivial E] (x : E) (r : ℝ) :
frontier (closed_ball x r) = sphere x r | by rw [frontier, closure_closed_ball, interior_closed_ball' x r, closed_ball_diff_ball] | theorem | frontier_closed_ball' | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"frontier",
"interior_closed_ball'",
"nontrivial",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_sphere' [normed_space ℝ E] [nontrivial E] (x : E) (r : ℝ) :
interior (sphere x r) = ∅ | by rw [←frontier_closed_ball' x, interior_frontier is_closed_ball] | theorem | interior_sphere' | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"interior",
"interior_frontier",
"nontrivial",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_sphere' [normed_space ℝ E] [nontrivial E] (x : E) (r : ℝ) :
frontier (sphere x r) = sphere x r | by rw [is_closed_sphere.frontier_eq, interior_sphere' x, diff_empty] | theorem | frontier_sphere' | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"frontier",
"interior_sphere'",
"nontrivial",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rescale_to_shell_zpow {c : α} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E} (hx : x ≠ 0) :
∃ n : ℤ, c ^ n ≠ 0 ∧ ‖c ^ n • x‖ < ε ∧ (ε / ‖c‖ ≤ ‖c ^ n • x‖) ∧ (‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖) | rescale_to_shell_semi_normed_zpow hc εpos (mt norm_eq_zero.1 hx) | lemma | rescale_to_shell_zpow | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"rescale_to_shell_semi_normed_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rescale_to_shell {c : α} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E} (hx : x ≠ 0) :
∃d:α, d ≠ 0 ∧ ‖d • x‖ < ε ∧ (ε/‖c‖ ≤ ‖d • x‖) ∧ (‖d‖⁻¹ ≤ ε⁻¹ * ‖c‖ * ‖x‖) | rescale_to_shell_semi_normed hc εpos (mt norm_eq_zero.1 hx) | lemma | rescale_to_shell | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"rescale_to_shell_semi_normed"
] | If there is a scalar `c` with `‖c‖>1`, then any element can be moved by scalar multiplication to
any shell of width `‖c‖`. Also recap information on the norm of the rescaling element that shows
up in applications. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_space.exists_lt_norm (c : ℝ) : ∃ x : E, c < ‖x‖ | begin
rcases exists_ne (0 : E) with ⟨x, hx⟩,
rcases normed_field.exists_lt_norm 𝕜 (c / ‖x‖) with ⟨r, hr⟩,
use r • x,
rwa [norm_smul, ← div_lt_iff],
rwa norm_pos_iff
end | lemma | normed_space.exists_lt_norm | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"div_lt_iff",
"exists_ne",
"norm_smul",
"normed_field.exists_lt_norm"
] | If `E` is a nontrivial normed space over a nontrivially normed field `𝕜`, then `E` is unbounded:
for any `c : ℝ`, there exists a vector `x : E` with norm strictly greater than `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_space.unbounded_univ : ¬bounded (univ : set E) | λ h, let ⟨R, hR⟩ := bounded_iff_forall_norm_le.1 h, ⟨x, hx⟩ := normed_space.exists_lt_norm 𝕜 E R
in hx.not_le (hR x trivial) | lemma | normed_space.unbounded_univ | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"normed_space.exists_lt_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_space.noncompact_space : noncompact_space E | ⟨λ h, normed_space.unbounded_univ 𝕜 _ h.bounded⟩ | lemma | normed_space.noncompact_space | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"noncompact_space",
"normed_space.unbounded_univ"
] | A normed vector space over a nontrivially normed field is a noncompact space. This cannot be
an instance because in order to apply it, Lean would have to search for `normed_space 𝕜 E` with
unknown `𝕜`. We register this as an instance in two cases: `𝕜 = E` and `𝕜 = ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nontrivially_normed_field.noncompact_space : noncompact_space 𝕜 | normed_space.noncompact_space 𝕜 𝕜 | instance | nontrivially_normed_field.noncompact_space | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"noncompact_space",
"normed_space.noncompact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real_normed_space.noncompact_space [normed_space ℝ E] : noncompact_space E | normed_space.noncompact_space ℝ E | instance | real_normed_space.noncompact_space | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"noncompact_space",
"normed_space",
"normed_space.noncompact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_algebra (𝕜 : Type*) (𝕜' : Type*) [normed_field 𝕜] [semi_normed_ring 𝕜']
extends algebra 𝕜 𝕜' | (norm_smul_le : ∀ (r : 𝕜) (x : 𝕜'), ‖r • x‖ ≤ ‖r‖ * ‖x‖) | class | normed_algebra | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"algebra",
"norm_smul_le",
"normed_field",
"semi_normed_ring"
] | A normed algebra `𝕜'` over `𝕜` is normed module that is also an algebra.
See the implementation notes for `algebra` for a discussion about non-unital algebras. Following
the strategy there, a non-unital *normed* algebra can be written as:
```lean
variables [normed_field 𝕜] [non_unital_semi_normed_ring 𝕜']
variable... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_algebra.to_normed_space : normed_space 𝕜 𝕜' | { norm_smul_le := normed_algebra.norm_smul_le } | instance | normed_algebra.to_normed_space | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"norm_smul_le",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_algebra.to_normed_space' {𝕜'} [normed_ring 𝕜'] [normed_algebra 𝕜 𝕜'] :
normed_space 𝕜 𝕜' | by apply_instance | instance | normed_algebra.to_normed_space' | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"normed_algebra",
"normed_ring",
"normed_space"
] | While this may appear identical to `normed_algebra.to_normed_space`, it contains an implicit
argument involving `normed_ring.to_semi_normed_ring` that typeclass inference has trouble inferring.
Specifically, the following instance cannot be found without this `normed_space.to_module'`:
```lean
example
(𝕜 ι : Type*)... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_algebra_map (x : 𝕜) : ‖algebra_map 𝕜 𝕜' x‖ = ‖x‖ * ‖(1 : 𝕜')‖ | begin
rw algebra.algebra_map_eq_smul_one,
exact norm_smul _ _,
end | lemma | norm_algebra_map | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"algebra.algebra_map_eq_smul_one",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_algebra_map (x : 𝕜) : ‖algebra_map 𝕜 𝕜' x‖₊ = ‖x‖₊ * ‖(1 : 𝕜')‖₊ | subtype.ext $ norm_algebra_map 𝕜' x | lemma | nnnorm_algebra_map | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"norm_algebra_map",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_algebra_map' [norm_one_class 𝕜'] (x : 𝕜) : ‖algebra_map 𝕜 𝕜' x‖ = ‖x‖ | by rw [norm_algebra_map, norm_one, mul_one] | lemma | norm_algebra_map' | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"mul_one",
"norm_algebra_map",
"norm_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_algebra_map' [norm_one_class 𝕜'] (x : 𝕜) : ‖algebra_map 𝕜 𝕜' x‖₊ = ‖x‖₊ | subtype.ext $ norm_algebra_map' _ _ | lemma | nnnorm_algebra_map' | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"norm_algebra_map'",
"norm_one_class",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_algebra_map_nnreal (x : ℝ≥0) : ‖algebra_map ℝ≥0 𝕜' x‖ = x | (norm_algebra_map' 𝕜' (x : ℝ)).symm ▸ real.norm_of_nonneg x.prop | lemma | norm_algebra_map_nnreal | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"norm_algebra_map'",
"real.norm_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_algebra_map_nnreal (x : ℝ≥0) : ‖algebra_map ℝ≥0 𝕜' x‖₊ = x | subtype.ext $ norm_algebra_map_nnreal 𝕜' x | lemma | nnnorm_algebra_map_nnreal | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"norm_algebra_map_nnreal",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_isometry [norm_one_class 𝕜'] : isometry (algebra_map 𝕜 𝕜') | begin
refine isometry.of_dist_eq (λx y, _),
rw [dist_eq_norm, dist_eq_norm, ← ring_hom.map_sub, norm_algebra_map'],
end | lemma | algebra_map_isometry | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"algebra_map",
"isometry",
"norm_algebra_map'",
"norm_one_class",
"ring_hom.map_sub"
] | In a normed algebra, the inclusion of the base field in the extended field is an isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_algebra.id : normed_algebra 𝕜 𝕜 | { .. normed_field.to_normed_space,
.. algebra.id 𝕜} | instance | normed_algebra.id | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"algebra.id",
"normed_algebra",
"normed_field.to_normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_algebra_rat {𝕜} [normed_division_ring 𝕜] [char_zero 𝕜] [normed_algebra ℝ 𝕜] :
normed_algebra ℚ 𝕜 | { norm_smul_le := λ q x,
by rw [←smul_one_smul ℝ q x, rat.smul_one_eq_coe, norm_smul, rat.norm_cast_real], } | instance | normed_algebra_rat | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"char_zero",
"norm_smul",
"norm_smul_le",
"normed_algebra",
"normed_division_ring",
"rat.norm_cast_real",
"rat.smul_one_eq_coe"
] | Any normed characteristic-zero division ring that is a normed_algebra over the reals is also a
normed algebra over the rationals.
Phrased another way, if `𝕜` is a normed algebra over the reals, then `algebra_rat` respects that
norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit.normed_algebra : normed_algebra 𝕜 punit | { norm_smul_le := λ q x, by simp only [punit.norm_eq_zero, mul_zero] } | instance | punit.normed_algebra | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"mul_zero",
"norm_smul_le",
"normed_algebra",
"punit.norm_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.normed_algebra {E F : Type*} [semi_normed_ring E] [semi_normed_ring F]
[normed_algebra 𝕜 E] [normed_algebra 𝕜 F] :
normed_algebra 𝕜 (E × F) | { ..prod.normed_space } | instance | prod.normed_algebra | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"normed_algebra",
"prod.normed_space",
"semi_normed_ring"
] | The product of two normed algebras is a normed algebra, with the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.normed_algebra {E : ι → Type*} [fintype ι]
[Π i, semi_normed_ring (E i)] [Π i, normed_algebra 𝕜 (E i)] :
normed_algebra 𝕜 (Π i, E i) | { .. pi.normed_space,
.. pi.algebra _ E } | instance | pi.normed_algebra | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"fintype",
"normed_algebra",
"pi.algebra",
"pi.normed_space",
"semi_normed_ring"
] | The product of finitely many normed algebras is a normed algebra, with the sup norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_opposite.normed_algebra {E : Type*} [semi_normed_ring E] [normed_algebra 𝕜 E] :
normed_algebra 𝕜 Eᵐᵒᵖ | { ..mul_opposite.normed_space } | instance | mul_opposite.normed_algebra | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"mul_opposite.normed_space",
"normed_algebra",
"semi_normed_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_algebra.induced {F : Type*} (α β γ : Type*) [normed_field α] [ring β]
[algebra α β] [semi_normed_ring γ] [normed_algebra α γ] [non_unital_alg_hom_class F α β γ]
(f : F) : @normed_algebra α β _ (semi_normed_ring.induced β γ f) | { norm_smul_le := λ a b, by {unfold norm, exact (map_smul f a b).symm ▸ norm_smul_le a (f b) } } | def | normed_algebra.induced | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"algebra",
"non_unital_alg_hom_class",
"norm_smul_le",
"normed_algebra",
"normed_field",
"ring",
"semi_normed_ring",
"semi_normed_ring.induced"
] | A non-unital algebra homomorphism from an `algebra` to a `normed_algebra` induces a
`normed_algebra` structure on the domain, using the `semi_normed_ring.induced` norm.
See note [reducible non-instances] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subalgebra.to_normed_algebra {𝕜 A : Type*} [semi_normed_ring A] [normed_field 𝕜]
[normed_algebra 𝕜 A] (S : subalgebra 𝕜 A) : normed_algebra 𝕜 S | @normed_algebra.induced _ 𝕜 S A _ (subring_class.to_ring S) S.algebra _ _ _ S.val | instance | subalgebra.to_normed_algebra | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"normed_algebra",
"normed_algebra.induced",
"normed_field",
"semi_normed_ring",
"subalgebra",
"subring_class.to_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.restrict_scalars.normed_space_orig {𝕜 : Type*} {𝕜' : Type*} {E : Type*}
[normed_field 𝕜'] [seminormed_add_comm_group E] [I : normed_space 𝕜' E] :
normed_space 𝕜' (restrict_scalars 𝕜 𝕜' E) | I | def | module.restrict_scalars.normed_space_orig | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"normed_field",
"normed_space",
"restrict_scalars",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_space.restrict_scalars : normed_space 𝕜 E | restrict_scalars.normed_space _ 𝕜' _ | def | normed_space.restrict_scalars | analysis.normed_space | src/analysis/normed_space/basic.lean | [
"algebra.algebra.pi",
"algebra.algebra.restrict_scalars",
"analysis.normed.field.basic",
"analysis.normed.mul_action",
"data.real.sqrt",
"topology.algebra.module.basic"
] | [
"normed_space"
] | Warning: This declaration should be used judiciously.
Please consider using `is_scalar_tower` and/or `restrict_scalars 𝕜 𝕜' E` instead.
This definition allows the `restrict_scalars.normed_space` instance to be put directly on `E`
rather on `restrict_scalars 𝕜 𝕜' E`. This would be a very bad instance; both because ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_linear_map (𝕜 : Type*) [normed_field 𝕜]
{E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E]
{F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F] (f : E → F)
extends is_linear_map 𝕜 f : Prop | (bound : ∃ M, 0 < M ∧ ∀ x : E, ‖f x‖ ≤ M * ‖x‖) | structure | is_bounded_linear_map | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"bound",
"is_linear_map",
"normed_add_comm_group",
"normed_field",
"normed_space"
] | A function `f` satisfies `is_bounded_linear_map 𝕜 f` if it is linear and satisfies the
inequality `‖f x‖ ≤ M * ‖x‖` for some positive constant `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_linear_map.with_bound
{f : E → F} (hf : is_linear_map 𝕜 f) (M : ℝ) (h : ∀ x : E, ‖f x‖ ≤ M * ‖x‖) :
is_bounded_linear_map 𝕜 f | ⟨ hf, classical.by_cases
(assume : M ≤ 0, ⟨1, zero_lt_one, λ x,
(h x).trans $ mul_le_mul_of_nonneg_right (this.trans zero_le_one) (norm_nonneg x)⟩)
(assume : ¬ M ≤ 0, ⟨M, lt_of_not_ge this, h⟩)⟩ | lemma | is_linear_map.with_bound | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map",
"is_linear_map",
"mul_le_mul_of_nonneg_right",
"zero_le_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.is_bounded_linear_map (f : E →L[𝕜] F) : is_bounded_linear_map 𝕜 f | { bound := f.bound,
..f.to_linear_map.is_linear } | lemma | continuous_linear_map.is_bounded_linear_map | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"bound",
"is_bounded_linear_map"
] | A continuous linear map satisfies `is_bounded_linear_map` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_map (f : E → F) (h : is_bounded_linear_map 𝕜 f) : E →ₗ[𝕜] F | (is_linear_map.mk' _ h.to_is_linear_map) | def | is_bounded_linear_map.to_linear_map | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map",
"is_linear_map.mk'"
] | Construct a linear map from a function `f` satisfying `is_bounded_linear_map 𝕜 f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_linear_map {f : E → F} (hf : is_bounded_linear_map 𝕜 f) : E →L[𝕜] F | { cont := let ⟨C, Cpos, hC⟩ :=
hf.bound in add_monoid_hom_class.continuous_of_bound (to_linear_map f hf) C hC,
..to_linear_map f hf} | def | is_bounded_linear_map.to_continuous_linear_map | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"cont",
"is_bounded_linear_map"
] | Construct a continuous linear map from is_bounded_linear_map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero : is_bounded_linear_map 𝕜 (λ (x:E), (0:F)) | (0 : E →ₗ[𝕜] F).is_linear.with_bound 0 $ by simp [le_refl] | lemma | is_bounded_linear_map.zero | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : is_bounded_linear_map 𝕜 (λ (x:E), x) | linear_map.id.is_linear.with_bound 1 $ by simp [le_refl] | lemma | is_bounded_linear_map.id | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst : is_bounded_linear_map 𝕜 (λ x : E × F, x.1) | begin
refine (linear_map.fst 𝕜 E F).is_linear.with_bound 1 (λ x, _),
rw one_mul,
exact le_max_left _ _
end | lemma | is_bounded_linear_map.fst | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map",
"linear_map.fst",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd : is_bounded_linear_map 𝕜 (λ x : E × F, x.2) | begin
refine (linear_map.snd 𝕜 E F).is_linear.with_bound 1 (λ x, _),
rw one_mul,
exact le_max_right _ _
end | lemma | is_bounded_linear_map.snd | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map",
"linear_map.snd",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul (c : 𝕜) (hf : is_bounded_linear_map 𝕜 f) :
is_bounded_linear_map 𝕜 (c • f) | let ⟨hlf, M, hMp, hM⟩ := hf in
(c • hlf.mk' f).is_linear.with_bound (‖c‖ * M) $ λ x,
calc ‖c • f x‖ = ‖c‖ * ‖f x‖ : norm_smul c (f x)
... ≤ ‖c‖ * (M * ‖x‖) : mul_le_mul_of_nonneg_left (hM _) (norm_nonneg _)
... = (‖c‖ * M) * ‖x‖ : (mul_assoc _ _ _).symm | lemma | is_bounded_linear_map.smul | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map",
"mul_assoc",
"mul_le_mul_of_nonneg_left",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg (hf : is_bounded_linear_map 𝕜 f) :
is_bounded_linear_map 𝕜 (λ e, -f e) | begin
rw show (λ e, -f e) = (λ e, (-1 : 𝕜) • f e), { funext, simp },
exact smul (-1) hf
end | lemma | is_bounded_linear_map.neg | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add (hf : is_bounded_linear_map 𝕜 f) (hg : is_bounded_linear_map 𝕜 g) :
is_bounded_linear_map 𝕜 (λ e, f e + g e) | let ⟨hlf, Mf, hMfp, hMf⟩ := hf in
let ⟨hlg, Mg, hMgp, hMg⟩ := hg in
(hlf.mk' _ + hlg.mk' _).is_linear.with_bound (Mf + Mg) $ λ x,
calc ‖f x + g x‖ ≤ Mf * ‖x‖ + Mg * ‖x‖ : norm_add_le_of_le (hMf x) (hMg x)
... ≤ (Mf + Mg) * ‖x‖ : by rw add_mul | lemma | is_bounded_linear_map.add | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub (hf : is_bounded_linear_map 𝕜 f) (hg : is_bounded_linear_map 𝕜 g) :
is_bounded_linear_map 𝕜 (λ e, f e - g e) | by simpa [sub_eq_add_neg] using add hf (neg hg) | lemma | is_bounded_linear_map.sub | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g : F → G}
(hg : is_bounded_linear_map 𝕜 g) (hf : is_bounded_linear_map 𝕜 f) :
is_bounded_linear_map 𝕜 (g ∘ f) | (hg.to_continuous_linear_map.comp hf.to_continuous_linear_map).is_bounded_linear_map | lemma | is_bounded_linear_map.comp | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto (x : E) (hf : is_bounded_linear_map 𝕜 f) :
tendsto f (𝓝 x) (𝓝 (f x)) | let ⟨hf, M, hMp, hM⟩ := hf in
tendsto_iff_norm_tendsto_zero.2 $
squeeze_zero (λ e, norm_nonneg _)
(λ e,
calc ‖f e - f x‖ = ‖hf.mk' f (e - x)‖ : by rw (hf.mk' _).map_sub e x; refl
... ≤ M * ‖e - x‖ : hM (e - x))
(suffices tendsto (λ (e : E), M * ‖e - x‖) (𝓝 x) (𝓝 (M * 0)), by ... | lemma | is_bounded_linear_map.tendsto | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map",
"squeeze_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (hf : is_bounded_linear_map 𝕜 f) : continuous f | continuous_iff_continuous_at.2 $ λ _, hf.tendsto _ | lemma | is_bounded_linear_map.continuous | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lim_zero_bounded_linear_map (hf : is_bounded_linear_map 𝕜 f) :
tendsto f (𝓝 0) (𝓝 0) | (hf.1.mk' _).map_zero ▸ continuous_iff_continuous_at.1 hf.continuous 0 | lemma | is_bounded_linear_map.lim_zero_bounded_linear_map | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_id {f : E → F} (h : is_bounded_linear_map 𝕜 f) (l : filter E) :
f =O[l] (λ x, x) | let ⟨M, hMp, hM⟩ := h.bound in is_O.of_bound _ (mem_of_superset univ_mem (λ x _, hM x)) | theorem | is_bounded_linear_map.is_O_id | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_comp {E : Type*} {g : F → G} (hg : is_bounded_linear_map 𝕜 g)
{f : E → F} (l : filter E) : (λ x', g (f x')) =O[l] f | (hg.is_O_id ⊤).comp_tendsto le_top | theorem | is_bounded_linear_map.is_O_comp | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"is_bounded_linear_map",
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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