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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comp (hg : is_conformal_map g) (hf : is_conformal_map f) :
is_conformal_map (g.comp f) | begin
rcases hf with ⟨cf, hcf, lif, rfl⟩,
rcases hg with ⟨cg, hcg, lig, rfl⟩,
refine ⟨cg * cf, mul_ne_zero hcg hcf, lig.comp lif, _⟩,
rw [smul_comp, comp_smul, mul_smul],
refl
end | lemma | is_conformal_map.comp | analysis.normed_space | src/analysis/normed_space/conformal_linear_map.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry"
] | [
"is_conformal_map",
"mul_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective {f : M' →L[R] N} (h : is_conformal_map f) : function.injective f | by { rcases h with ⟨c, hc, li, rfl⟩, exact (smul_right_injective _ hc).comp li.injective } | lemma | is_conformal_map.injective | analysis.normed_space | src/analysis/normed_space/conformal_linear_map.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry"
] | [
"is_conformal_map",
"smul_right_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero [nontrivial M'] {f' : M' →L[R] N} (hf' : is_conformal_map f') :
f' ≠ 0 | begin
rintro rfl,
rcases exists_ne (0 : M') with ⟨a, ha⟩,
exact ha (hf'.injective rfl)
end | lemma | is_conformal_map.ne_zero | analysis.normed_space | src/analysis/normed_space/conformal_linear_map.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry"
] | [
"exists_ne",
"is_conformal_map",
"ne_zero",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_linear (f : P →A[R] Q) : V →L[R] W | { to_fun := f.linear,
cont := by { rw affine_map.continuous_linear_iff, exact f.cont, },
.. f.linear, } | def | continuous_affine_map.cont_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [
"affine_map.continuous_linear_iff",
"cont"
] | The linear map underlying a continuous affine map is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_cont_linear (f : P →A[R] Q) :
(f.cont_linear : V → W) = f.linear | rfl | lemma | continuous_affine_map.coe_cont_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_cont_linear_eq_linear (f : P →A[R] Q) :
(f.cont_linear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear | by { ext, refl, } | lemma | continuous_affine_map.coe_cont_linear_eq_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).cont_linear : V → W) = f.linear | rfl | lemma | continuous_affine_map.coe_mk_const_linear_eq_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_linear_eq_coe_cont_linear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.cont_linear : V → W) | rfl | lemma | continuous_affine_map.coe_linear_eq_coe_cont_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_cont_linear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).cont_linear = g.cont_linear.comp f.cont_linear | rfl | lemma | continuous_affine_map.comp_cont_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_vadd (f : P →A[R] Q) (p : P) (v : V) :
f (v +ᵥ p) = f.cont_linear v +ᵥ f p | f.map_vadd' p v | lemma | continuous_affine_map.map_vadd | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_linear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) :
f.cont_linear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ | f.to_affine_map.linear_map_vsub p₁ p₂ | lemma | continuous_affine_map.cont_linear_map_vsub | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_cont_linear (q : Q) : (const R P q).cont_linear = 0 | rfl | lemma | continuous_affine_map.const_cont_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_linear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.cont_linear = 0 ↔ ∃ q, f = const R P q | begin
have h₁ : f.cont_linear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0,
{ refine ⟨λ h, _, λ h, _⟩;
ext,
{ rw [← coe_cont_linear_eq_linear, h], refl, },
{ rw [← coe_linear_eq_coe_cont_linear, h], refl, }, },
have h₂ : ∀ (q : Q), f = const R P q ↔ (f : P →ᵃ[R] Q) = affine_map.const R P q,
{ intros q,
refi... | lemma | continuous_affine_map.cont_linear_eq_zero_iff_exists_const | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [
"affine_map.const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_affine_map_cont_linear (f : V →L[R] W) :
f.to_continuous_affine_map.cont_linear = f | by { ext, refl, } | lemma | continuous_affine_map.to_affine_map_cont_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_cont_linear :
(0 : P →A[R] W).cont_linear = 0 | rfl | lemma | continuous_affine_map.zero_cont_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_cont_linear (f g : P →A[R] W) :
(f + g).cont_linear = f.cont_linear + g.cont_linear | rfl | lemma | continuous_affine_map.add_cont_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_cont_linear (f g : P →A[R] W) :
(f - g).cont_linear = f.cont_linear - g.cont_linear | rfl | lemma | continuous_affine_map.sub_cont_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_cont_linear (f : P →A[R] W) :
(-f).cont_linear = -f.cont_linear | rfl | lemma | continuous_affine_map.neg_cont_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_cont_linear (t : R) (f : P →A[R] W) :
(t • f).cont_linear = t • f.cont_linear | rfl | lemma | continuous_affine_map.smul_cont_linear | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomp (f : V →A[R] W) :
(f : V → W) = f.cont_linear + function.const V (f 0) | begin
rcases f with ⟨f, h⟩,
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, pi.add_apply, linear_map.map_zero, zero_add],
end | lemma | continuous_affine_map.decomp | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [
"linear_map.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_norm : has_norm (V →A[𝕜] W) | ⟨λ f, max ‖f 0‖ ‖f.cont_linear‖⟩ | instance | continuous_affine_map.has_norm | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [
"has_norm"
] | Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_def : ‖f‖ = (max ‖f 0‖ ‖f.cont_linear‖) | rfl | lemma | continuous_affine_map.norm_def | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_cont_linear_le : ‖f.cont_linear‖ ≤ ‖f‖ | le_max_right _ _ | lemma | continuous_affine_map.norm_cont_linear_le | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ | le_max_left _ _ | lemma | continuous_affine_map.norm_image_zero_le | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.cont_linear‖ | calc ‖f‖ = (max ‖f 0‖ ‖f.cont_linear‖) : by rw norm_def
... = (max 0 ‖f.cont_linear‖) : by rw [h, norm_zero]
... = ‖f.cont_linear‖ : max_eq_right (norm_nonneg _) | lemma | continuous_affine_map.norm_eq | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_comp_le (g : W₂ →A[𝕜] V) :
‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ | begin
rw [norm_def, max_le_iff],
split,
{ calc ‖f.comp g 0‖ = ‖f (g 0)‖ : by simp
... = ‖f.cont_linear (g 0) + f 0‖ : by { rw f.decomp, simp, }
... ≤ ‖f.cont_linear‖ * ‖g 0‖ + ‖f 0‖ :
(norm_add_le _ _).trans (add_le_add_right (f.cont_linear.le_op_norm _)... | lemma | continuous_affine_map.norm_comp_le | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [
"max_le_iff",
"mul_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_const_prod_continuous_linear_map : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) | { to_fun := λ f, ⟨f 0, f.cont_linear⟩,
inv_fun := λ p, p.2.to_continuous_affine_map + const 𝕜 V p.1,
left_inv := λ f, by { ext, rw f.decomp, simp, },
right_inv := by { rintros ⟨v, f⟩, ext; simp, },
map_add' := λ _ _, rfl,
map_smul' := λ _ _, rfl,
norm_map' := λ f, rfl } | def | continuous_affine_map.to_const_prod_continuous_linear_map | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [
"inv_fun"
] | The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_const_prod_continuous_linear_map_fst (f : V →A[𝕜] W) :
(to_const_prod_continuous_linear_map 𝕜 V W f).fst = f 0 | rfl | lemma | continuous_affine_map.to_const_prod_continuous_linear_map_fst | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_const_prod_continuous_linear_map_snd (f : V →A[𝕜] W) :
(to_const_prod_continuous_linear_map 𝕜 V W f).snd = f.cont_linear | rfl | lemma | continuous_affine_map.to_const_prod_continuous_linear_map_snd | analysis.normed_space | src/analysis/normed_space/continuous_affine_map.lean | [
"topology.algebra.continuous_affine_map",
"analysis.normed_space.affine_isometry",
"analysis.normed_space.operator_norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.mk_continuous (C : ℝ) (h : ∀x, ‖f x‖ ≤ C * ‖x‖) : E →SL[σ] F | ⟨f, add_monoid_hom_class.continuous_of_bound f C h⟩ | def | linear_map.mk_continuous | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | Construct a continuous linear map from a linear map and a bound on this linear map.
The fact that the norm of the continuous linear map is then controlled is given in
`linear_map.mk_continuous_norm_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.mk_continuous_of_exists_bound (h : ∃C, ∀x, ‖f x‖ ≤ C * ‖x‖) : E →SL[σ] F | ⟨f, let ⟨C, hC⟩ := h in add_monoid_hom_class.continuous_of_bound f C hC⟩ | def | linear_map.mk_continuous_of_exists_bound | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | Construct a continuous linear map from a linear map and the existence of a bound on this linear
map. If you have an explicit bound, use `linear_map.mk_continuous` instead, as a norm estimate will
follow automatically in `linear_map.mk_continuous_norm_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_of_linear_of_boundₛₗ {f : E → F} (h_add : ∀ x y, f (x + y) = f x + f y)
(h_smul : ∀ (c : 𝕜) x, f (c • x) = (σ c) • f x) {C : ℝ} (h_bound : ∀ x, ‖f x‖ ≤ C*‖x‖) :
continuous f | let φ : E →ₛₗ[σ] F := { to_fun := f, map_add' := h_add, map_smul' := h_smul } in
add_monoid_hom_class.continuous_of_bound φ C h_bound | lemma | continuous_of_linear_of_boundₛₗ | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_linear_of_bound {f : E → G} (h_add : ∀ x y, f (x + y) = f x + f y)
(h_smul : ∀ (c : 𝕜) x, f (c • x) = c • f x) {C : ℝ} (h_bound : ∀ x, ‖f x‖ ≤ C*‖x‖) :
continuous f | let φ : E →ₗ[𝕜] G := { to_fun := f, map_add' := h_add, map_smul' := h_smul } in
add_monoid_hom_class.continuous_of_bound φ C h_bound | lemma | continuous_of_linear_of_bound | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.mk_continuous_coe (C : ℝ) (h : ∀x, ‖f x‖ ≤ C * ‖x‖) :
((f.mk_continuous C h) : E →ₛₗ[σ] F) = f | rfl | lemma | linear_map.mk_continuous_coe | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.mk_continuous_apply (C : ℝ) (h : ∀x, ‖f x‖ ≤ C * ‖x‖) (x : E) :
f.mk_continuous C h x = f x | rfl | lemma | linear_map.mk_continuous_apply | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.mk_continuous_of_exists_bound_coe
(h : ∃C, ∀x, ‖f x‖ ≤ C * ‖x‖) :
((f.mk_continuous_of_exists_bound h) : E →ₛₗ[σ] F) = f | rfl | lemma | linear_map.mk_continuous_of_exists_bound_coe | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.mk_continuous_of_exists_bound_apply (h : ∃C, ∀x, ‖f x‖ ≤ C * ‖x‖) (x : E) :
f.mk_continuous_of_exists_bound h x = f x | rfl | lemma | linear_map.mk_continuous_of_exists_bound_apply | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz_of_bound (f : E →SL[σ] F) {K : ℝ≥0} (h : ∀ x, ‖x‖ ≤ K * ‖f x‖) :
antilipschitz_with K f | add_monoid_hom_class.antilipschitz_of_bound _ h | theorem | continuous_linear_map.antilipschitz_of_bound | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bound_of_antilipschitz (f : E →SL[σ] F) {K : ℝ≥0} (h : antilipschitz_with K f) (x) :
‖x‖ ≤ K * ‖f x‖ | zero_hom_class.bound_of_antilipschitz _ h x | lemma | continuous_linear_map.bound_of_antilipschitz | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv.to_continuous_linear_equiv_of_bounds (e : E ≃ₛₗ[σ] F) (C_to C_inv : ℝ)
(h_to : ∀ x, ‖e x‖ ≤ C_to * ‖x‖) (h_inv : ∀ x : F, ‖e.symm x‖ ≤ C_inv * ‖x‖) : E ≃SL[σ] F | { to_linear_equiv := e,
continuous_to_fun := add_monoid_hom_class.continuous_of_bound e C_to h_to,
continuous_inv_fun := add_monoid_hom_class.continuous_of_bound e.symm C_inv h_inv } | def | linear_equiv.to_continuous_linear_equiv_of_bounds | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | Construct a continuous linear equivalence from a linear equivalence together with
bounds in both directions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.to_continuous_linear_map₁ (f : 𝕜 →ₗ[𝕜] E) : 𝕜 →L[𝕜] E | f.mk_continuous (‖f 1‖) $ λ x,
by { conv_lhs { rw ← mul_one x }, rw [← smul_eq_mul, f.map_smul, mul_comm],exact norm_smul_le _ _ } | def | linear_map.to_continuous_linear_map₁ | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [
"mul_comm",
"mul_one",
"norm_smul_le",
"smul_eq_mul"
] | Reinterpret a linear map `𝕜 →ₗ[𝕜] E` as a continuous linear map. This construction
is generalized to the case of any finite dimensional domain
in `linear_map.to_continuous_linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map.to_continuous_linear_map₁_coe (f : 𝕜 →ₗ[𝕜] E) :
(f.to_continuous_linear_map₁ : 𝕜 →ₗ[𝕜] E) = f | rfl | lemma | linear_map.to_continuous_linear_map₁_coe | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map.to_continuous_linear_map₁_apply (f : 𝕜 →ₗ[𝕜] E) (x) :
f.to_continuous_linear_map₁ x = f x | rfl | lemma | linear_map.to_continuous_linear_map₁_apply | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.uniform_embedding_of_bound {K : ℝ≥0} (hf : ∀ x, ‖x‖ ≤ K * ‖f x‖) :
uniform_embedding f | (add_monoid_hom_class.antilipschitz_of_bound f hf).uniform_embedding f.uniform_continuous | theorem | continuous_linear_map.uniform_embedding_of_bound | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.of_homothety (f : E →ₛₗ[σ] F) (a : ℝ) (hf : ∀x, ‖f x‖ = a * ‖x‖) :
E →SL[σ] F | f.mk_continuous a (λ x, le_of_eq (hf x)) | def | continuous_linear_map.of_homothety | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | A (semi-)linear map which is a homothety is a continuous linear map.
Since the field `𝕜` need not have `ℝ` as a subfield, this theorem is not directly deducible from
the corresponding theorem about isometries plus a theorem about scalar multiplication. Likewise
for the other theorems about homotheties in ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.homothety_inverse (a : ℝ) (ha : 0 < a) (f : E ≃ₛₗ[σ] F) :
(∀ (x : E), ‖f x‖ = a * ‖x‖) → (∀ (y : F), ‖f.symm y‖ = a⁻¹ * ‖y‖) | begin
intros hf y,
calc ‖(f.symm) y‖ = a⁻¹ * (a * ‖ (f.symm) y‖) : _
... = a⁻¹ * ‖f ((f.symm) y)‖ : by rw hf
... = a⁻¹ * ‖y‖ : by simp,
rw [← mul_assoc, inv_mul_cancel (ne_of_lt ha).symm, one_mul],
end | lemma | continuous_linear_equiv.homothety_inverse | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [
"inv_mul_cancel",
"mul_assoc",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_equiv.of_homothety (f : E ≃ₛₗ[σ] F) (a : ℝ) (ha : 0 < a)
(hf : ∀x, ‖f x‖ = a * ‖x‖) :
E ≃SL[σ] F | linear_equiv.to_continuous_linear_equiv_of_bounds f a a⁻¹
(λ x, (hf x).le) (λ x, (continuous_linear_equiv.homothety_inverse a ha f hf x).le) | def | continuous_linear_equiv.of_homothety | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [
"continuous_linear_equiv.homothety_inverse",
"linear_equiv.to_continuous_linear_equiv_of_bounds"
] | A linear equivalence which is a homothety is a continuous linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_span_singleton_homothety (x : E) (c : 𝕜) :
‖linear_map.to_span_singleton 𝕜 E x c‖ = ‖x‖ * ‖c‖ | by {rw mul_comm, exact norm_smul _ _} | lemma | continuous_linear_map.to_span_singleton_homothety | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [
"mul_comm",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_span_nonzero_singleton_homothety (x : E) (h : x ≠ 0) (c : 𝕜) :
‖linear_equiv.to_span_nonzero_singleton 𝕜 E x h c‖ = ‖x‖ * ‖c‖ | continuous_linear_map.to_span_singleton_homothety _ _ _ | lemma | continuous_linear_equiv.to_span_nonzero_singleton_homothety | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [
"continuous_linear_map.to_span_singleton_homothety"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_span_nonzero_singleton (x : E) (h : x ≠ 0) : 𝕜 ≃L[𝕜] (𝕜 ∙ x) | of_homothety
(linear_equiv.to_span_nonzero_singleton 𝕜 E x h)
‖x‖
(norm_pos_iff.mpr h)
(to_span_nonzero_singleton_homothety 𝕜 x h) | def | continuous_linear_equiv.to_span_nonzero_singleton | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [
"linear_equiv.to_span_nonzero_singleton"
] | Given a nonzero element `x` of a normed space `E₁` over a field `𝕜`, the natural
continuous linear equivalence from `E₁` to the span of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coord (x : E) (h : x ≠ 0) : (𝕜 ∙ x) →L[𝕜] 𝕜 | (to_span_nonzero_singleton 𝕜 x h).symm | def | continuous_linear_equiv.coord | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | Given a nonzero element `x` of a normed space `E₁` over a field `𝕜`, the natural continuous
linear map from the span of `x` to `𝕜`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_span_nonzero_singleton_symm {x : E} (h : x ≠ 0) :
⇑(to_span_nonzero_singleton 𝕜 x h).symm = coord 𝕜 x h | rfl | lemma | continuous_linear_equiv.coe_to_span_nonzero_singleton_symm | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_to_span_nonzero_singleton {x : E} (h : x ≠ 0) (c : 𝕜) :
coord 𝕜 x h (to_span_nonzero_singleton 𝕜 x h c) = c | (to_span_nonzero_singleton 𝕜 x h).symm_apply_apply c | lemma | continuous_linear_equiv.coord_to_span_nonzero_singleton | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_span_nonzero_singleton_coord {x : E} (h : x ≠ 0) (y : 𝕜 ∙ x) :
to_span_nonzero_singleton 𝕜 x h (coord 𝕜 x h y) = y | (to_span_nonzero_singleton 𝕜 x h).apply_symm_apply y | lemma | continuous_linear_equiv.to_span_nonzero_singleton_coord | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_self (x : E) (h : x ≠ 0) :
(coord 𝕜 x h) (⟨x, submodule.mem_span_singleton_self x⟩ : 𝕜 ∙ x) = 1 | linear_equiv.coord_self 𝕜 E x h | lemma | continuous_linear_equiv.coord_self | analysis.normed_space | src/analysis/normed_space/continuous_linear_map.lean | [
"analysis.normed_space.basic"
] | [
"linear_equiv.coord_self",
"submodule.mem_span_singleton_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual | E →L[𝕜] 𝕜 | def | normed_space.dual | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [] | The topological dual of a seminormed space `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion_in_double_dual : E →L[𝕜] (dual 𝕜 (dual 𝕜 E)) | continuous_linear_map.apply 𝕜 𝕜 | def | normed_space.inclusion_in_double_dual | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"continuous_linear_map.apply"
] | The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_def (x : E) (f : dual 𝕜 E) : inclusion_in_double_dual 𝕜 E x f = f x | rfl | lemma | normed_space.dual_def | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_in_double_dual_norm_eq :
‖inclusion_in_double_dual 𝕜 E‖ = ‖(continuous_linear_map.id 𝕜 (dual 𝕜 E))‖ | continuous_linear_map.op_norm_flip _ | lemma | normed_space.inclusion_in_double_dual_norm_eq | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"continuous_linear_map.id",
"continuous_linear_map.op_norm_flip"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_in_double_dual_norm_le : ‖inclusion_in_double_dual 𝕜 E‖ ≤ 1 | by { rw inclusion_in_double_dual_norm_eq, exact continuous_linear_map.norm_id_le } | lemma | normed_space.inclusion_in_double_dual_norm_le | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"continuous_linear_map.norm_id_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
double_dual_bound (x : E) : ‖(inclusion_in_double_dual 𝕜 E) x‖ ≤ ‖x‖ | by simpa using continuous_linear_map.le_of_op_norm_le _ (inclusion_in_double_dual_norm_le 𝕜 E) x | lemma | normed_space.double_dual_bound | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"continuous_linear_map.le_of_op_norm_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_pairing : (dual 𝕜 E) →ₗ[𝕜] E →ₗ[𝕜] 𝕜 | continuous_linear_map.coe_lm 𝕜 | def | normed_space.dual_pairing | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"continuous_linear_map.coe_lm"
] | The dual pairing as a bilinear form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_pairing_apply {v : dual 𝕜 E} {x : E} : dual_pairing 𝕜 E v x = v x | rfl | lemma | normed_space.dual_pairing_apply | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"dual_pairing_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_pairing_separating_left : (dual_pairing 𝕜 E).separating_left | begin
rw [linear_map.separating_left_iff_ker_eq_bot, linear_map.ker_eq_bot],
exact continuous_linear_map.coe_injective,
end | lemma | normed_space.dual_pairing_separating_left | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"continuous_linear_map.coe_injective",
"linear_map.ker_eq_bot",
"linear_map.separating_left_iff_ker_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_le_dual_bound (x : E) {M : ℝ} (hMp: 0 ≤ M) (hM : ∀ (f : dual 𝕜 E), ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M | begin
classical,
by_cases h : x = 0,
{ simp only [h, hMp, norm_zero] },
{ obtain ⟨f, hf₁, hfx⟩ : ∃ f : E →L[𝕜] 𝕜, ‖f‖ = 1 ∧ f x = ‖x‖ := exists_dual_vector 𝕜 x h,
calc ‖x‖ = ‖(‖x‖ : 𝕜)‖ : is_R_or_C.norm_coe_norm.symm
... = ‖f x‖ : by rw hfx
... ≤ M * ‖f‖ : hM f
... = M : by rw [hf₁, mul_one]... | lemma | normed_space.norm_le_dual_bound | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"exists_dual_vector",
"mul_one"
] | If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `continuous_linear_map.op_norm_le_bound`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_of_forall_dual_eq_zero {x : E} (h : ∀ f : dual 𝕜 E, f x = (0 : 𝕜)) : x = 0 | norm_le_zero_iff.mp (norm_le_dual_bound 𝕜 x le_rfl (λ f, by simp [h f])) | lemma | normed_space.eq_zero_of_forall_dual_eq_zero | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 ↔ ∀ g : dual 𝕜 E, g x = 0 | ⟨λ hx, by simp [hx], λ h, eq_zero_of_forall_dual_eq_zero 𝕜 h⟩ | lemma | normed_space.eq_zero_iff_forall_dual_eq_zero | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_iff_forall_dual_eq {x y : E} :
x = y ↔ ∀ g : dual 𝕜 E, g x = g y | begin
rw [← sub_eq_zero, eq_zero_iff_forall_dual_eq_zero 𝕜 (x - y)],
simp [sub_eq_zero],
end | lemma | normed_space.eq_iff_forall_dual_eq | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [] | See also `geometric_hahn_banach_point_point`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion_in_double_dual_li : E →ₗᵢ[𝕜] (dual 𝕜 (dual 𝕜 E)) | { norm_map' := begin
intros x,
apply le_antisymm,
{ exact double_dual_bound 𝕜 E x },
rw continuous_linear_map.norm_def,
refine le_cInf continuous_linear_map.bounds_nonempty _,
rintros c ⟨hc1, hc2⟩,
exact norm_le_dual_bound 𝕜 x hc1 hc2
end,
.. inclusion_in_double_dual 𝕜 E } | def | normed_space.inclusion_in_double_dual_li | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"continuous_linear_map.bounds_nonempty",
"continuous_linear_map.norm_def",
"le_cInf"
] | The inclusion of a normed space in its double dual is an isometry onto its image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polar (𝕜 : Type*) [nontrivially_normed_field 𝕜]
{E : Type*} [seminormed_add_comm_group E] [normed_space 𝕜 E] : set E → set (dual 𝕜 E) | (dual_pairing 𝕜 E).flip.polar | def | normed_space.polar | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"nontrivially_normed_field",
"normed_space",
"seminormed_add_comm_group"
] | Given a subset `s` in a normed space `E` (over a field `𝕜`), the polar
`polar 𝕜 s` is the subset of `dual 𝕜 E` consisting of those functionals which
evaluate to something of norm at most one at all points `z ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_polar_iff {x' : dual 𝕜 E} (s : set E) : x' ∈ polar 𝕜 s ↔ ∀ z ∈ s, ‖x' z‖ ≤ 1 | iff.rfl | lemma | normed_space.mem_polar_iff | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar_univ : polar 𝕜 (univ : set E) = {(0 : dual 𝕜 E)} | (dual_pairing 𝕜 E).flip.polar_univ
(linear_map.flip_separating_right.mpr (dual_pairing_separating_left 𝕜 E)) | lemma | normed_space.polar_univ | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_polar (s : set E) : is_closed (polar 𝕜 s) | begin
dunfold normed_space.polar,
simp only [linear_map.polar_eq_Inter, linear_map.flip_apply],
refine is_closed_bInter (λ z hz, _),
exact is_closed_Iic.preimage (continuous_linear_map.apply 𝕜 𝕜 z).continuous.norm
end | lemma | normed_space.is_closed_polar | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"continuous_linear_map.apply",
"is_closed",
"is_closed_bInter",
"linear_map.flip_apply",
"linear_map.polar_eq_Inter",
"normed_space.polar"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar_closure (s : set E) : polar 𝕜 (closure s) = polar 𝕜 s | ((dual_pairing 𝕜 E).flip.polar_antitone subset_closure).antisymm $
(dual_pairing 𝕜 E).flip.polar_gc.l_le $
closure_minimal ((dual_pairing 𝕜 E).flip.polar_gc.le_u_l s) $
by simpa [linear_map.flip_flip]
using (is_closed_polar _ _).preimage (inclusion_in_double_dual 𝕜 E).continuous | lemma | normed_space.polar_closure | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"closure",
"closure_minimal",
"continuous",
"linear_map.flip_flip",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_polar {s : set E} {x' : dual 𝕜 E} {c : 𝕜}
(hc : ∀ z, z ∈ s → ‖ x' z ‖ ≤ ‖c‖) : c⁻¹ • x' ∈ polar 𝕜 s | begin
by_cases c_zero : c = 0, { simp only [c_zero, inv_zero, zero_smul],
exact (dual_pairing 𝕜 E).flip.zero_mem_polar _ },
have eq : ∀ z, ‖ c⁻¹ • (x' z) ‖ = ‖ c⁻¹ ‖ * ‖ x' z ‖ := λ z, norm_smul c⁻¹ _,
have le : ∀ z, z ∈ s → ‖ c⁻¹ • (x' z) ‖ ≤ ‖ c⁻¹ ‖ * ‖ c ‖,
{ intros z hzs,
rw eq z,
apply mul_le_... | lemma | normed_space.smul_mem_polar | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"inv_mul_cancel",
"inv_zero",
"mul_le_mul",
"norm_eq_zero",
"norm_inv",
"norm_smul",
"zero_smul"
] | If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polar_ball_subset_closed_ball_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closed_ball (0 : dual 𝕜 E) (‖c‖ / r) | begin
intros x' hx',
rw mem_polar_iff at hx',
simp only [polar, mem_set_of_eq, mem_closed_ball_zero_iff, mem_ball_zero_iff] at *,
have hcr : 0 < ‖c‖ / r, from div_pos (zero_lt_one.trans hc) hr,
refine continuous_linear_map.op_norm_le_of_shell hr hcr.le hc (λ x h₁ h₂, _),
calc ‖x' x‖ ≤ 1 : hx' _ h₂
... ≤ (... | lemma | normed_space.polar_ball_subset_closed_ball_div | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"continuous_linear_map.op_norm_le_of_shell",
"div_pos",
"inv_div",
"inv_pos_le_iff_one_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_inv_subset_polar_closed_ball {r : ℝ} :
closed_ball (0 : dual 𝕜 E) r⁻¹ ⊆ polar 𝕜 (closed_ball (0 : E) r) | λ x' hx' x hx,
calc ‖x' x‖ ≤ ‖x'‖ * ‖x‖ : x'.le_op_norm x
... ≤ r⁻¹ * r :
mul_le_mul (mem_closed_ball_zero_iff.1 hx') (mem_closed_ball_zero_iff.1 hx)
(norm_nonneg _) (dist_nonneg.trans hx')
... = r / r : inv_mul_eq_div _ _
... ≤ 1 : div_self_le_one r | lemma | normed_space.closed_ball_inv_subset_polar_closed_ball | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"div_self_le_one",
"inv_mul_eq_div",
"mul_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar_closed_ball {𝕜 E : Type*} [is_R_or_C 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E]
{r : ℝ} (hr : 0 < r) :
polar 𝕜 (closed_ball (0 : E) r) = closed_ball (0 : dual 𝕜 E) r⁻¹ | begin
refine subset.antisymm _ (closed_ball_inv_subset_polar_closed_ball _),
intros x' h,
simp only [mem_closed_ball_zero_iff],
refine continuous_linear_map.op_norm_le_of_ball hr (inv_nonneg.mpr hr.le) (λ z hz, _),
simpa only [one_div] using linear_map.bound_of_ball_bound' hr 1 x'.to_linear_map h z
end | lemma | normed_space.polar_closed_ball | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"continuous_linear_map.op_norm_le_of_ball",
"is_R_or_C",
"linear_map.bound_of_ball_bound'",
"normed_add_comm_group",
"normed_space",
"one_div"
] | The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_polar_of_mem_nhds_zero {s : set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
bounded (polar 𝕜 s) | begin
obtain ⟨a, ha⟩ : ∃ a : 𝕜, 1 < ‖a‖ := normed_field.exists_one_lt_norm 𝕜,
obtain ⟨r, r_pos, r_ball⟩ : ∃ (r : ℝ) (hr : 0 < r), ball 0 r ⊆ s :=
metric.mem_nhds_iff.1 s_nhd,
exact bounded_closed_ball.mono (((dual_pairing 𝕜 E).flip.polar_antitone r_ball).trans $
polar_ball_subset_closed_ball_div ha r_p... | lemma | normed_space.bounded_polar_of_mem_nhds_zero | analysis.normed_space | src/analysis/normed_space/dual.lean | [
"analysis.normed_space.hahn_banach.extension",
"analysis.normed_space.is_R_or_C",
"analysis.locally_convex.polar"
] | [
"normed_field.exists_one_lt_norm"
] | Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
of all elements of the polar `polar 𝕜 s` are bounded by a constant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_eps : exp 𝕜 (eps : dual_number R) = 1 + eps | exp_inr _ _ | lemma | dual_number.exp_eps | analysis.normed_space | src/analysis/normed_space/dual_number.lean | [
"algebra.dual_number",
"analysis.normed_space.triv_sq_zero_ext"
] | [
"dual_number",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_smul_eps (r : R) : exp 𝕜 (r • eps : dual_number R) = 1 + r • eps | by rw [eps, ←inr_smul, exp_inr] | lemma | dual_number.exp_smul_eps | analysis.normed_space | src/analysis/normed_space/dual_number.lean | [
"algebra.dual_number",
"analysis.normed_space.triv_sq_zero_ext"
] | [
"dual_number",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enorm (𝕜 : Type*) (V : Type*) [normed_field 𝕜] [add_comm_group V] [module 𝕜 V] | (to_fun : V → ℝ≥0∞)
(eq_zero' : ∀ x, to_fun x = 0 → x = 0)
(map_add_le' : ∀ x y : V, to_fun (x + y) ≤ to_fun x + to_fun y)
(map_smul_le' : ∀ (c : 𝕜) (x : V), to_fun (c • x) ≤ ‖c‖₊ * to_fun x) | structure | enorm | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"add_comm_group",
"module",
"normed_field"
] | Extended norm on a vector space. As in the case of normed spaces, we require only
`‖c • x‖ ≤ ‖c‖ * ‖x‖` in the definition, then prove an equality in `map_smul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_injective : function.injective (coe_fn : enorm 𝕜 V → (V → ℝ≥0∞)) | λ e₁ e₂ h, by cases e₁; cases e₂; congr; exact h | lemma | enorm.coe_fn_injective | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"enorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {e₁ e₂ : enorm 𝕜 V} (h : ∀ x, e₁ x = e₂ x) : e₁ = e₂ | coe_fn_injective $ funext h | lemma | enorm.ext | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"enorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {e₁ e₂ : enorm 𝕜 V} : e₁ = e₂ ↔ ∀ x, e₁ x = e₂ x | ⟨λ h x, h ▸ rfl, ext⟩ | lemma | enorm.ext_iff | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"enorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj {e₁ e₂ : enorm 𝕜 V} : (e₁ : V → ℝ≥0∞) = e₂ ↔ e₁ = e₂ | coe_fn_injective.eq_iff | lemma | enorm.coe_inj | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"enorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul (c : 𝕜) (x : V) : e (c • x) = ‖c‖₊ * e x | le_antisymm (e.map_smul_le' c x) $
begin
by_cases hc : c = 0, { simp [hc] },
calc (‖c‖₊ : ℝ≥0∞) * e x = ‖c‖₊ * e (c⁻¹ • c • x) : by rw [inv_smul_smul₀ hc]
... ≤ ‖c‖₊ * (‖c⁻¹‖₊ * e (c • x)) : _
... = e (c • x) : _,
{ exact mul_le_mul_left' (e.map_smul_le' _ _) _ },
{ rw [← mul_assoc, nnnorm_inv, ennreal.coe_... | lemma | enorm.map_smul | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"ennreal.coe_inv",
"ennreal.coe_ne_top",
"ennreal.mul_inv_cancel",
"inv_smul_smul₀",
"mul_assoc",
"mul_le_mul_left'",
"nnnorm_inv",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero : e 0 = 0 | by { rw [← zero_smul 𝕜 (0:V), e.map_smul], norm_num } | lemma | enorm.map_zero | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_iff {x : V} : e x = 0 ↔ x = 0 | ⟨e.eq_zero' x, λ h, h.symm ▸ e.map_zero⟩ | lemma | enorm.eq_zero_iff | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg (x : V) : e (-x) = e x | calc e (-x) = ‖(-1 : 𝕜)‖₊ * e x : by rw [← map_smul, neg_one_smul]
... = e x : by simp | lemma | enorm.map_neg | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"neg_one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub_rev (x y : V) : e (x - y) = e (y - x) | by rw [← neg_sub, e.map_neg] | lemma | enorm.map_sub_rev | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add_le (x y : V) : e (x + y) ≤ e x + e y | e.map_add_le' x y | lemma | enorm.map_add_le | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub_le (x y : V) : e (x - y) ≤ e x + e y | calc e (x - y) = e (x + -y) : by rw sub_eq_add_neg
... ≤ e x + e (-y) : e.map_add_le x (-y)
... = e x + e y : by rw [e.map_neg] | lemma | enorm.map_sub_le | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_map {x : V} (hx : x ≠ 0) : (⊤ : enorm 𝕜 V) x = ⊤ | if_neg hx | lemma | enorm.top_map | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"enorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_max (e₁ e₂ : enorm 𝕜 V) : ⇑(e₁ ⊔ e₂) = λ x, max (e₁ x) (e₂ x) | rfl | lemma | enorm.coe_max | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"enorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_map (e₁ e₂ : enorm 𝕜 V) (x : V) : (e₁ ⊔ e₂) x = max (e₁ x) (e₂ x) | rfl | lemma | enorm.max_map | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"enorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emetric_space : emetric_space V | { edist := λ x y, e (x - y),
edist_self := λ x, by simp,
eq_of_edist_eq_zero := λ x y, by simp [sub_eq_zero],
edist_comm := e.map_sub_rev,
edist_triangle := λ x y z,
calc e (x - z) = e ((x - y) + (y - z)) : by rw [sub_add_sub_cancel]
... ≤ e (x - y) + e (y - z) : e.map_add_le (x - y) (y - z) ... | def | enorm.emetric_space | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"emetric_space"
] | Structure of an `emetric_space` defined by an extended norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_subspace : subspace 𝕜 V | { carrier := {x | e x < ⊤},
zero_mem' := by simp,
add_mem' := λ x y hx hy, lt_of_le_of_lt (e.map_add_le x y) (ennreal.add_lt_top.2 ⟨hx, hy⟩),
smul_mem' := λ c x (hx : _ < _),
calc e (c • x) = ‖c‖₊ * e x : e.map_smul c x
... < ⊤ : ennreal.mul_lt_top ennreal.coe_ne_top hx.ne } | def | enorm.finite_subspace | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [
"ennreal.coe_ne_top",
"ennreal.mul_lt_top",
"subspace"
] | The subspace of vectors with finite enorm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_dist_eq (x y : e.finite_subspace) : dist x y = (e (x - y)).to_real | rfl | lemma | enorm.finite_dist_eq | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_edist_eq (x y : e.finite_subspace) : edist x y = e (x - y) | rfl | lemma | enorm.finite_edist_eq | analysis.normed_space | src/analysis/normed_space/enorm.lean | [
"analysis.normed_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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