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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|---|---|---|---|---|---|---|---|---|---|---|
_root_.continuous.prod_map_equivL {f : X → M₁ ≃L[𝕜] M₂} {g : X → M₃ ≃L[𝕜] M₄}
(hf : continuous (λ x, (f x : M₁ →L[𝕜] M₂))) (hg : continuous (λ x, (g x : M₃ →L[𝕜] M₄))) :
continuous (λ x, ((f x).prod (g x) : M₁ × M₃ →L[𝕜] M₂ × M₄)) | (prod_mapL 𝕜 M₁ M₂ M₃ M₄).continuous.comp (hf.prod_mk hg) | lemma | continuous.prod_map_equivL | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous",
"continuous.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_on.prod_mapL {f : X → M₁ →L[𝕜] M₂} {g : X → M₃ →L[𝕜] M₄} {s : set X}
(hf : continuous_on f s) (hg : continuous_on g s) :
continuous_on (λ x, (f x).prod_map (g x)) s | ((prod_mapL 𝕜 M₁ M₂ M₃ M₄).continuous.comp_continuous_on (hf.prod hg) : _) | lemma | continuous_on.prod_mapL | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous.comp_continuous_on",
"continuous_on",
"prod_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_on.prod_map_equivL {f : X → M₁ ≃L[𝕜] M₂} {g : X → M₃ ≃L[𝕜] M₄} {s : set X}
(hf : continuous_on (λ x, (f x : M₁ →L[𝕜] M₂)) s)
(hg : continuous_on (λ x, (g x : M₃ →L[𝕜] M₄)) s) :
continuous_on (λ x, ((f x).prod (g x) : M₁ × M₃ →L[𝕜] M₂ × M₄)) s | (prod_mapL 𝕜 M₁ M₂ M₃ M₄).continuous.comp_continuous_on (hf.prod hg) | lemma | continuous_on.prod_map_equivL | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous.comp_continuous_on",
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul : 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' | (linear_map.mul 𝕜 𝕜').mk_continuous₂ 1 $
λ x y, by simpa using norm_mul_le x y | def | continuous_linear_map.mul | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"linear_map.mul",
"norm_mul_le"
] | Multiplication in a non-unital normed algebra as a continuous bilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_apply' (x y : 𝕜') : mul 𝕜 𝕜' x y = x * y | rfl | lemma | continuous_linear_map.mul_apply' | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_mul_apply_le (x : 𝕜') : ‖mul 𝕜 𝕜' x‖ ≤ ‖x‖ | (op_norm_le_bound _ (norm_nonneg x) (norm_mul_le x)) | lemma | continuous_linear_map.op_norm_mul_apply_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"norm_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_mul_le : ‖mul 𝕜 𝕜'‖ ≤ 1 | linear_map.mk_continuous₂_norm_le _ zero_le_one _ | lemma | continuous_linear_map.op_norm_mul_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"linear_map.mk_continuous₂_norm_le",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_right : 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' | ((compL 𝕜 𝕜' 𝕜' 𝕜').comp (mul 𝕜 𝕜').flip).flip.comp (mul 𝕜 𝕜') | def | continuous_linear_map.mul_left_right | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | Simultaneous left- and right-multiplication in a non-unital normed algebra, considered as a
continuous trilinear map. This is akin to its non-continuous version `linear_map.mul_left_right`,
but there is a minor difference: `linear_map.mul_left_right` is uncurried. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left_right_apply (x y z : 𝕜') :
mul_left_right 𝕜 𝕜' x y z = x * z * y | rfl | lemma | continuous_linear_map.mul_left_right_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_mul_left_right_apply_apply_le (x y : 𝕜') :
‖mul_left_right 𝕜 𝕜' x y‖ ≤ ‖x‖ * ‖y‖ | (op_norm_comp_le _ _).trans $ (mul_comm _ _).trans_le $
mul_le_mul (op_norm_mul_apply_le _ _ _)
(op_norm_le_bound _ (norm_nonneg _) (λ _, (norm_mul_le _ _).trans_eq (mul_comm _ _)))
(norm_nonneg _) (norm_nonneg _) | lemma | continuous_linear_map.op_norm_mul_left_right_apply_apply_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"mul_comm",
"mul_le_mul",
"norm_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_mul_left_right_apply_le (x : 𝕜') :
‖mul_left_right 𝕜 𝕜' x‖ ≤ ‖x‖ | op_norm_le_bound _ (norm_nonneg x) (op_norm_mul_left_right_apply_apply_le 𝕜 𝕜' x) | lemma | continuous_linear_map.op_norm_mul_left_right_apply_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_mul_left_right_le :
‖mul_left_right 𝕜 𝕜'‖ ≤ 1 | op_norm_le_bound _ zero_le_one (λ x, (one_mul ‖x‖).symm ▸ op_norm_mul_left_right_apply_le 𝕜 𝕜' x) | lemma | continuous_linear_map.op_norm_mul_left_right_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"one_mul",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mulₗᵢ : 𝕜' →ₗᵢ[𝕜] 𝕜' →L[𝕜] 𝕜' | { to_linear_map := mul 𝕜 𝕜',
norm_map' := λ x, le_antisymm (op_norm_mul_apply_le _ _ _)
(by { convert ratio_le_op_norm _ (1 : 𝕜'), simp [norm_one],
apply_instance }) } | def | continuous_linear_map.mulₗᵢ | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | Multiplication in a normed algebra as a linear isometry to the space of
continuous linear maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mulₗᵢ : ⇑(mulₗᵢ 𝕜 𝕜') = mul 𝕜 𝕜' | rfl | lemma | continuous_linear_map.coe_mulₗᵢ | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_mul_apply (x : 𝕜') : ‖mul 𝕜 𝕜' x‖ = ‖x‖ | (mulₗᵢ 𝕜 𝕜').norm_map x | lemma | continuous_linear_map.op_norm_mul_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsmul : 𝕜' →L[𝕜] E →L[𝕜] E | ((algebra.lsmul 𝕜 E).to_linear_map : 𝕜' →ₗ[𝕜] E →ₗ[𝕜] E).mk_continuous₂ 1 $
λ c x, by simpa only [one_mul] using norm_smul_le c x | def | continuous_linear_map.lsmul | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"algebra.lsmul",
"norm_smul_le",
"one_mul"
] | Scalar multiplication as a continuous bilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lsmul_apply (c : 𝕜') (x : E) : lsmul 𝕜 𝕜' c x = c • x | rfl | lemma | continuous_linear_map.lsmul_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_to_span_singleton (x : E) : ‖to_span_singleton 𝕜 x‖ = ‖x‖ | begin
refine op_norm_eq_of_bounds (norm_nonneg _) (λ x, _) (λ N hN_nonneg h, _),
{ rw [to_span_singleton_apply, norm_smul, mul_comm], },
{ specialize h 1,
rw [to_span_singleton_apply, norm_smul, mul_comm] at h,
exact (mul_le_mul_right (by simp)).mp h, },
end | lemma | continuous_linear_map.norm_to_span_singleton | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"mul_comm",
"mul_le_mul_right",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_lsmul_apply_le (x : 𝕜') : ‖(lsmul 𝕜 𝕜' x : E →L[𝕜] E)‖ ≤ ‖x‖ | continuous_linear_map.op_norm_le_bound _ (norm_nonneg x) $ λ y, norm_smul_le x y | lemma | continuous_linear_map.op_norm_lsmul_apply_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.op_norm_le_bound",
"norm_smul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_lsmul_le : ‖(lsmul 𝕜 𝕜' : 𝕜' →L[𝕜] E →L[𝕜] E)‖ ≤ 1 | begin
refine continuous_linear_map.op_norm_le_bound _ zero_le_one (λ x, _),
simp_rw [one_mul],
exact op_norm_lsmul_apply_le _,
end | lemma | continuous_linear_map.op_norm_lsmul_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.op_norm_le_bound",
"one_mul",
"zero_le_one"
] | The norm of `lsmul` is at most 1 in any semi-normed group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_restrict_scalars (f : E →L[𝕜] Fₗ) : ‖f.restrict_scalars 𝕜'‖ = ‖f‖ | le_antisymm (op_norm_le_bound _ (norm_nonneg _) $ λ x, f.le_op_norm x)
(op_norm_le_bound _ (norm_nonneg _) $ λ x, f.le_op_norm x) | lemma | continuous_linear_map.norm_restrict_scalars | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_isometry : (E →L[𝕜] Fₗ) →ₗᵢ[𝕜''] (E →L[𝕜'] Fₗ) | ⟨restrict_scalarsₗ 𝕜 E Fₗ 𝕜' 𝕜'', norm_restrict_scalars⟩ | def | continuous_linear_map.restrict_scalars_isometry | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | `continuous_linear_map.restrict_scalars` as a `linear_isometry`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_restrict_scalars_isometry :
⇑(restrict_scalars_isometry 𝕜 E Fₗ 𝕜' 𝕜'') = restrict_scalars 𝕜' | rfl | lemma | continuous_linear_map.coe_restrict_scalars_isometry | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_isometry_to_linear_map :
(restrict_scalars_isometry 𝕜 E Fₗ 𝕜' 𝕜'').to_linear_map = restrict_scalarsₗ 𝕜 E Fₗ 𝕜' 𝕜'' | rfl | lemma | continuous_linear_map.restrict_scalars_isometry_to_linear_map | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalarsL : (E →L[𝕜] Fₗ) →L[𝕜''] (E →L[𝕜'] Fₗ) | (restrict_scalars_isometry 𝕜 E Fₗ 𝕜' 𝕜'').to_continuous_linear_map | def | continuous_linear_map.restrict_scalarsL | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | `continuous_linear_map.restrict_scalars` as a `continuous_linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_restrict_scalarsL :
(restrict_scalarsL 𝕜 E Fₗ 𝕜' 𝕜'' : (E →L[𝕜] Fₗ) →ₗ[𝕜''] (E →L[𝕜'] Fₗ)) =
restrict_scalarsₗ 𝕜 E Fₗ 𝕜' 𝕜'' | rfl | lemma | continuous_linear_map.coe_restrict_scalarsL | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_restrict_scalarsL' :
⇑(restrict_scalarsL 𝕜 E Fₗ 𝕜' 𝕜'') = restrict_scalars 𝕜' | rfl | lemma | continuous_linear_map.coe_restrict_scalarsL' | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_subtypeL_le (K : submodule 𝕜 E) : ‖K.subtypeL‖ ≤ 1 | K.subtypeₗᵢ.norm_to_continuous_linear_map_le | lemma | submodule.norm_subtypeL_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz : lipschitz_with (‖(e : E →SL[σ₁₂] F)‖₊) e | (e : E →SL[σ₁₂] F).lipschitz | lemma | continuous_linear_equiv.lipschitz | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_comp {α : Type*} (f : α → E) (l : filter α) : (λ x', e (f x')) =O[l] f | (e : E →SL[σ₁₂] F).is_O_comp f l | theorem | continuous_linear_equiv.is_O_comp | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_sub (l : filter E) (x : E) : (λ x', e (x' - x)) =O[l] (λ x', x' - x) | (e : E →SL[σ₁₂] F).is_O_sub l x | theorem | continuous_linear_equiv.is_O_sub | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_comp_rev {α : Type*} (f : α → E) (l : filter α) : f =O[l] (λ x', e (f x')) | (e.symm.is_O_comp _ l).congr_left $ λ _, e.symm_apply_apply _ | theorem | continuous_linear_equiv.is_O_comp_rev | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_sub_rev (l : filter E) (x : E) : (λ x', x' - x) =O[l] (λ x', e (x' - x)) | e.is_O_comp_rev _ _ | theorem | continuous_linear_equiv.is_O_sub_rev | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bilinear_comp (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (gE : E' →SL[σ₁'] E) (gF : F' →SL[σ₂'] F) :
E' →SL[σ₁₃'] F' →SL[σ₂₃'] G | ((f.comp gE).flip.comp gF).flip | def | continuous_linear_map.bilinear_comp | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | Compose a bilinear map `E →SL[σ₁₃] F →SL[σ₂₃] G` with two linear maps
`E' →SL[σ₁'] E` and `F' →SL[σ₂'] F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bilinear_comp_apply (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (gE : E' →SL[σ₁'] E)
(gF : F' →SL[σ₂'] F) (x : E') (y : F') : f.bilinear_comp gE gF x y = f (gE x) (gF y) | rfl | lemma | continuous_linear_map.bilinear_comp_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv₂ (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : (E × Fₗ) →L[𝕜] (E × Fₗ) →L[𝕜] Gₗ | f.bilinear_comp (fst _ _ _) (snd _ _ _) + f.flip.bilinear_comp (snd _ _ _) (fst _ _ _) | def | continuous_linear_map.deriv₂ | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | Derivative of a continuous bilinear map `f : E →L[𝕜] F →L[𝕜] G` interpreted as a map `E × F → G`
at point `p : E × F` evaluated at `q : E × F`, as a continuous bilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_deriv₂ (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (p : E × Fₗ) :
⇑(f.deriv₂ p) = λ q : E × Fₗ, f p.1 q.2 + f q.1 p.2 | rfl | lemma | continuous_linear_map.coe_deriv₂ | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add_add (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (x x' : E) (y y' : Fₗ) :
f (x + x') (y + y') = f x y + f.deriv₂ (x, y) (x', y') + f x' y' | by simp only [map_add, add_apply, coe_deriv₂, add_assoc] | lemma | continuous_linear_map.map_add_add | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bound_of_shell [ring_hom_isometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) {ε C : ℝ} (ε_pos : 0 < ε) {c : 𝕜}
(hc : 1 < ‖c‖) (hf : ∀ x, ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) (x : E) :
‖f x‖ ≤ C * ‖x‖ | begin
by_cases hx : x = 0, { simp [hx] },
exact semilinear_map_class.bound_of_shell_semi_normed f ε_pos hc hf
(ne_of_lt (norm_pos_iff.2 hx)).symm
end | lemma | linear_map.bound_of_shell | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"ring_hom_isometric",
"semilinear_map_class.bound_of_shell_semi_normed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bound_of_ball_bound {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] Fₗ)
(h : ∀ z ∈ metric.ball (0 : E) r, ‖f z‖ ≤ c) :
∃ C, ∀ (z : E), ‖f z‖ ≤ C * ‖z‖ | begin
cases @nontrivially_normed_field.non_trivial 𝕜 _ with k hk,
use c * (‖k‖ / r),
intro z,
refine bound_of_shell _ r_pos hk (λ x hko hxo, _) _,
calc ‖f x‖ ≤ c : h _ (mem_ball_zero_iff.mpr hxo)
... ≤ c * ((‖x‖ * ‖k‖) / r) : le_mul_of_one_le_right _ _
... = _ : by ring,
{ exact le_trans ... | lemma | linear_map.bound_of_ball_bound | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"div_le_iff",
"le_mul_of_one_le_right",
"metric.ball",
"one_le_div",
"ring"
] | `linear_map.bound_of_ball_bound'` is a version of this lemma over a field satisfying `is_R_or_C`
that produces a concrete bound. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antilipschitz_of_comap_nhds_le [h : ring_hom_isometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
(hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, antilipschitz_with K f | begin
rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos
with ⟨ε, ε0, hε⟩,
simp only [set.subset_def, set.mem_preimage, mem_ball_zero_iff] at hε,
lift ε to ℝ≥0 using ε0.le,
rcases normed_field.exists_one_lt_norm 𝕜 with ⟨c, hc⟩,
refine ⟨ε⁻¹ * ‖c‖₊, add_monoid_hom_class.antilips... | lemma | linear_map.antilipschitz_of_comap_nhds_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"antilipschitz_with",
"filter.tendsto_pure_pure",
"inv_smul_smul₀",
"lift",
"map_zpow₀",
"mul_le_mul_of_nonneg_left",
"mul_one",
"nnreal.coe_nonneg",
"norm_inv",
"norm_smul",
"normed_field.exists_one_lt_norm",
"pure_le_nhds",
"rescale_to_shell_zpow",
"ring_hom_isometric",
"set.mem_preima... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_zero_iff [ring_hom_isometric σ₁₂] : ‖f‖ = 0 ↔ f = 0 | iff.intro
(λ hn, continuous_linear_map.ext (λ x, norm_le_zero_iff.1
(calc _ ≤ ‖f‖ * ‖x‖ : le_op_norm _ _
... = _ : by rw [hn, zero_mul])))
(by { rintro rfl, exact op_norm_zero }) | theorem | continuous_linear_map.op_norm_zero_iff | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.ext",
"ring_hom_isometric",
"zero_mul"
] | An operator is zero iff its norm vanishes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_id [nontrivial E] : ‖id 𝕜 E‖ = 1 | begin
refine norm_id_of_nontrivial_seminorm _,
obtain ⟨x, hx⟩ := exists_ne (0 : E),
exact ⟨x, ne_of_gt (norm_pos_iff.2 hx)⟩,
end | lemma | continuous_linear_map.norm_id | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"exists_ne",
"nontrivial"
] | If a normed space is non-trivial, then the norm of the identity equals `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_one_class [nontrivial E] : norm_one_class (E →L[𝕜] E) | ⟨norm_id⟩ | instance | continuous_linear_map.norm_one_class | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"nontrivial",
"norm_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_normed_add_comm_group [ring_hom_isometric σ₁₂] : normed_add_comm_group (E →SL[σ₁₂] F) | normed_add_comm_group.of_separation (λ f, (op_norm_zero_iff f).mp) | instance | continuous_linear_map.to_normed_add_comm_group | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"normed_add_comm_group",
"ring_hom_isometric"
] | Continuous linear maps themselves form a normed space with respect to
the operator norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_normed_ring : normed_ring (E →L[𝕜] E) | { .. continuous_linear_map.to_normed_add_comm_group, .. continuous_linear_map.to_semi_normed_ring } | instance | continuous_linear_map.to_normed_ring | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.to_normed_add_comm_group",
"continuous_linear_map.to_semi_normed_ring",
"normed_ring"
] | Continuous linear maps form a normed ring with respect to the operator norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homothety_norm [ring_hom_isometric σ₁₂] [nontrivial E] (f : E →SL[σ₁₂] F) {a : ℝ}
(hf : ∀x, ‖f x‖ = a * ‖x‖) :
‖f‖ = a | begin
obtain ⟨x, hx⟩ : ∃ (x : E), x ≠ 0 := exists_ne 0,
rw ← norm_pos_iff at hx,
have ha : 0 ≤ a, by simpa only [hf, hx, zero_le_mul_right] using norm_nonneg (f x),
apply le_antisymm (f.op_norm_le_bound ha (λ y, le_of_eq (hf y))),
simpa only [hf, hx, mul_le_mul_right] using f.le_op_norm x,
end | lemma | continuous_linear_map.homothety_norm | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"exists_ne",
"mul_le_mul_right",
"nontrivial",
"ring_hom_isometric",
"zero_le_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz_of_embedding (f : E →L[𝕜] Fₗ) (hf : embedding f) :
∃ K, antilipschitz_with K f | f.to_linear_map.antilipschitz_of_comap_nhds_le $ map_zero f ▸ (hf.nhds_eq_comap 0).ge | theorem | continuous_linear_map.antilipschitz_of_embedding | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"antilipschitz_with",
"embedding"
] | If a continuous linear map is a topology embedding, then it is expands the distances
by a positive factor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_mem_closure_image_coe_bounded (f : E' → F) {s : set (E' →SL[σ₁₂] F)} (hs : bounded s)
(hf : f ∈ closure ((coe_fn : (E' →SL[σ₁₂] F) → E' → F) '' s)) :
E' →SL[σ₁₂] F | begin
-- `f` is a linear map due to `linear_map_of_mem_closure_range_coe`
refine (linear_map_of_mem_closure_range_coe f _).mk_continuous_of_exists_bound _,
{ refine closure_mono (image_subset_iff.2 $ λ g hg, _) hf, exact ⟨g, rfl⟩ },
{ -- We need to show that `f` has bounded norm. Choose `C` such that `‖g‖ ≤ C` ... | def | continuous_linear_map.of_mem_closure_image_coe_bounded | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"closure",
"closure_mono",
"continuous_apply",
"is_closed",
"linear_map_of_mem_closure_range_coe"
] | Construct a bundled continuous (semi)linear map from a map `f : E → F` and a proof of the fact
that it belongs to the closure of the image of a bounded set `s : set (E →SL[σ₁₂] F)` under coercion
to function. Coercion to function of the result is definitionally equal to `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_tendsto_of_bounded_range {α : Type*} {l : filter α} [l.ne_bot] (f : E' → F)
(g : α → E' →SL[σ₁₂] F) (hf : tendsto (λ a x, g a x) l (𝓝 f)) (hg : bounded (set.range g)) :
E' →SL[σ₁₂] F | of_mem_closure_image_coe_bounded f hg $ mem_closure_of_tendsto hf $
eventually_of_forall $ λ a, mem_image_of_mem _ $ set.mem_range_self _ | def | continuous_linear_map.of_tendsto_of_bounded_range | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"filter",
"mem_closure_of_tendsto",
"set.mem_range_self",
"set.range"
] | Let `f : E → F` be a map, let `g : α → E →SL[σ₁₂] F` be a family of continuous (semi)linear maps
that takes values in a bounded set and converges to `f` pointwise along a nontrivial filter. Then
`f` is a continuous (semi)linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_of_tendsto_pointwise_of_cauchy_seq {f : ℕ → E' →SL[σ₁₂] F} {g : E' →SL[σ₁₂] F}
(hg : tendsto (λ n x, f n x) at_top (𝓝 g)) (hf : cauchy_seq f) :
tendsto f at_top (𝓝 g) | begin
/- Since `f` is a Cauchy sequence, there exists `b → 0` such that `‖f n - f m‖ ≤ b N` for any
`m, n ≥ N`. -/
rcases cauchy_seq_iff_le_tendsto_0.1 hf with ⟨b, hb₀, hfb, hb_lim⟩,
-- Since `b → 0`, it suffices to show that `‖f n x - g x‖ ≤ b n * ‖x‖` for all `n` and `x`.
suffices : ∀ n x, ‖f n x - g x‖ ≤ b... | lemma | continuous_linear_map.tendsto_of_tendsto_pointwise_of_cauchy_seq | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"cauchy_seq",
"le_of_tendsto",
"le_rfl",
"squeeze_zero"
] | If a Cauchy sequence of continuous linear map converges to a continuous linear map pointwise,
then it converges to the same map in norm. This lemma is used to prove that the space of continuous
linear maps is complete provided that the codomain is a complete space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_closure_image_coe_of_bounded [proper_space F] {s : set (E' →SL[σ₁₂] F)}
(hb : bounded s) :
is_compact (closure ((coe_fn : (E' →SL[σ₁₂] F) → E' → F) '' s)) | have ∀ x, is_compact (closure (apply' F σ₁₂ x '' s)),
from λ x, ((apply' F σ₁₂ x).lipschitz.bounded_image hb).is_compact_closure,
is_compact_closure_of_subset_compact (is_compact_pi_infinite this)
(image_subset_iff.2 $ λ g hg x, subset_closure $ mem_image_of_mem _ hg) | lemma | continuous_linear_map.is_compact_closure_image_coe_of_bounded | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"closure",
"is_compact",
"is_compact_closure_of_subset_compact",
"is_compact_pi_infinite",
"proper_space",
"subset_closure"
] | Let `s` be a bounded set in the space of continuous (semi)linear maps `E →SL[σ] F` taking values
in a proper space. Then `s` interpreted as a set in the space of maps `E → F` with topology of
pointwise convergence is precompact: its closure is a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_image_coe_of_bounded_of_closed_image [proper_space F] {s : set (E' →SL[σ₁₂] F)}
(hb : bounded s) (hc : is_closed ((coe_fn : (E' →SL[σ₁₂] F) → E' → F) '' s)) :
is_compact ((coe_fn : (E' →SL[σ₁₂] F) → E' → F) '' s) | hc.closure_eq ▸ is_compact_closure_image_coe_of_bounded hb | lemma | continuous_linear_map.is_compact_image_coe_of_bounded_of_closed_image | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"is_closed",
"is_compact",
"proper_space"
] | Let `s` be a bounded set in the space of continuous (semi)linear maps `E →SL[σ] F` taking values
in a proper space. If `s` interpreted as a set in the space of maps `E → F` with topology of
pointwise convergence is closed, then it is compact.
TODO: reformulate this in terms of a type synonym with the right topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_image_coe_of_bounded_of_weak_closed {s : set (E' →SL[σ₁₂] F)} (hb : bounded s)
(hc : ∀ f, (⇑f : E' → F) ∈ closure ((coe_fn : (E' →SL[σ₁₂] F) → E' → F) '' s) → f ∈ s) :
is_closed ((coe_fn : (E' →SL[σ₁₂] F) → E' → F) '' s) | is_closed_of_closure_subset $ λ f hf,
⟨of_mem_closure_image_coe_bounded f hb hf, hc (of_mem_closure_image_coe_bounded f hb hf) hf, rfl⟩ | lemma | continuous_linear_map.is_closed_image_coe_of_bounded_of_weak_closed | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"closure",
"is_closed",
"is_closed_of_closure_subset"
] | If a set `s` of semilinear functions is bounded and is closed in the weak-* topology, then its
image under coercion to functions `E → F` is a closed set. We don't have a name for `E →SL[σ] F`
with weak-* topology in `mathlib`, so we use an equivalent condition (see `is_closed_induced_iff'`).
TODO: reformulate this in ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_image_coe_of_bounded_of_weak_closed [proper_space F] {s : set (E' →SL[σ₁₂] F)}
(hb : bounded s)
(hc : ∀ f, (⇑f : E' → F) ∈ closure ((coe_fn : (E' →SL[σ₁₂] F) → E' → F) '' s) → f ∈ s) :
is_compact ((coe_fn : (E' →SL[σ₁₂] F) → E' → F) '' s) | is_compact_image_coe_of_bounded_of_closed_image hb $
is_closed_image_coe_of_bounded_of_weak_closed hb hc | lemma | continuous_linear_map.is_compact_image_coe_of_bounded_of_weak_closed | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"closure",
"is_compact",
"proper_space"
] | If a set `s` of semilinear functions is bounded and is closed in the weak-* topology, then its
image under coercion to functions `E → F` is a compact set. We don't have a name for `E →SL[σ] F`
with weak-* topology in `mathlib`, so we use an equivalent condition (see `is_closed_induced_iff'`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_weak_closed_closed_ball (f₀ : E' →SL[σ₁₂] F) (r : ℝ) ⦃f : E' →SL[σ₁₂] F⦄
(hf : ⇑f ∈ closure ((coe_fn : (E' →SL[σ₁₂] F) → E' → F) '' (closed_ball f₀ r))) :
f ∈ closed_ball f₀ r | begin
have hr : 0 ≤ r,
from nonempty_closed_ball.1 (nonempty_image_iff.1 (closure_nonempty_iff.1 ⟨_, hf⟩)),
refine mem_closed_ball_iff_norm.2 (op_norm_le_bound _ hr $ λ x, _),
have : is_closed {g : E' → F | ‖g x - f₀ x‖ ≤ r * ‖x‖},
from is_closed_Iic.preimage ((@continuous_apply E' (λ _, F) _ x).sub conti... | lemma | continuous_linear_map.is_weak_closed_closed_ball | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"closure",
"continuous_apply",
"continuous_const",
"is_closed"
] | A closed ball is closed in the weak-* topology. We don't have a name for `E →SL[σ] F` with
weak-* topology in `mathlib`, so we use an equivalent condition (see `is_closed_induced_iff'`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_image_coe_closed_ball (f₀ : E →SL[σ₁₂] F) (r : ℝ) :
is_closed ((coe_fn : (E →SL[σ₁₂] F) → E → F) '' closed_ball f₀ r) | is_closed_image_coe_of_bounded_of_weak_closed bounded_closed_ball (is_weak_closed_closed_ball f₀ r) | lemma | continuous_linear_map.is_closed_image_coe_closed_ball | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"is_closed"
] | The set of functions `f : E → F` that represent continuous linear maps `f : E →SL[σ₁₂] F`
at distance `≤ r` from `f₀ : E →SL[σ₁₂] F` is closed in the topology of pointwise convergence.
This is one of the key steps in the proof of the **Banach-Alaoglu** theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_image_coe_closed_ball [proper_space F] (f₀ : E →SL[σ₁₂] F) (r : ℝ) :
is_compact ((coe_fn : (E →SL[σ₁₂] F) → E → F) '' closed_ball f₀ r) | is_compact_image_coe_of_bounded_of_weak_closed bounded_closed_ball $
is_weak_closed_closed_ball f₀ r | lemma | continuous_linear_map.is_compact_image_coe_closed_ball | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"is_compact",
"proper_space"
] | **Banach-Alaoglu** theorem. The set of functions `f : E → F` that represent continuous linear
maps `f : E →SL[σ₁₂] F` at distance `≤ r` from `f₀ : E →SL[σ₁₂] F` is compact in the topology of
pointwise convergence. Other versions of this theorem can be found in
`analysis.normed_space.weak_dual`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend : Fₗ →SL[σ₁₂] F | /- extension of `f` is continuous -/
have cont : _ := (uniform_continuous_uniformly_extend h_e h_dense f.uniform_continuous).continuous,
/- extension of `f` agrees with `f` on the domain of the embedding `e` -/
have eq : _ := uniformly_extend_of_ind h_e h_dense f.uniform_continuous,
{ to_fun := (h_e.dense_inducing h_de... | def | continuous_linear_map.extend | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"cont",
"continuous",
"continuous_fst",
"continuous_linear_map.map_smulₛₗ",
"continuous_snd",
"extend",
"is_closed_eq",
"uniform_continuous_uniformly_extend",
"uniformly_extend_of_ind"
] | Extension of a continuous linear map `f : E →SL[σ₁₂] F`, with `E` a normed space and `F` a
complete normed space, along a uniform and dense embedding `e : E →L[𝕜] Fₗ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_eq (x : E) : extend f e h_dense h_e (e x) = f x | dense_inducing.extend_eq _ f.cont _ | lemma | continuous_linear_map.extend_eq | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"dense_inducing.extend_eq",
"extend"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_unique (g : Fₗ →SL[σ₁₂] F) (H : g.comp e = f) : extend f e h_dense h_e = g | continuous_linear_map.coe_fn_injective $
uniformly_extend_unique h_e h_dense (continuous_linear_map.ext_iff.1 H) g.continuous | lemma | continuous_linear_map.extend_unique | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.coe_fn_injective",
"extend",
"uniformly_extend_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_zero : extend (0 : E →SL[σ₁₂] F) e h_dense h_e = 0 | extend_unique _ _ _ _ _ (zero_comp _) | lemma | continuous_linear_map.extend_zero | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"extend"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_extend_le : ‖ψ‖ ≤ N * ‖f‖ | begin
have uni : uniform_inducing e := (uniform_embedding_of_bound _ h_e).to_uniform_inducing,
have eq : ∀x, ψ (e x) = f x := uniformly_extend_of_ind uni h_dense f.uniform_continuous,
by_cases N0 : 0 ≤ N,
{ refine op_norm_le_bound ψ _ (is_closed_property h_dense (is_closed_le _ _) _),
{ exact mul_nonneg N0 ... | lemma | continuous_linear_map.op_norm_extend_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"cont",
"is_closed_le",
"is_closed_property",
"mul_assoc",
"mul_comm",
"mul_le_mul_of_nonneg_left",
"mul_nonpos_of_nonpos_of_nonneg",
"mul_zero",
"uniform_inducing",
"uniformly_extend_of_ind"
] | If a dense embedding `e : E →L[𝕜] G` expands the norm by a constant factor `N⁻¹`, then the
norm of the extension of `f` along `e` is bounded by `N * ‖f‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_to_continuous_linear_map [nontrivial E] [ring_hom_isometric σ₁₂]
(f : E →ₛₗᵢ[σ₁₂] F) :
‖f.to_continuous_linear_map‖ = 1 | f.to_continuous_linear_map.homothety_norm $ by simp | lemma | linear_isometry.norm_to_continuous_linear_map | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"nontrivial",
"ring_hom_isometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_to_continuous_linear_map_comp [ring_hom_isometric σ₁₂] (f : F →ₛₗᵢ[σ₂₃] G)
{g : E →SL[σ₁₂] F} :
‖f.to_continuous_linear_map.comp g‖ = ‖g‖ | op_norm_ext (f.to_continuous_linear_map.comp g) g
(λ x, by simp only [norm_map, coe_to_continuous_linear_map, coe_comp']) | lemma | linear_isometry.norm_to_continuous_linear_map_comp | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"ring_hom_isometric"
] | Postcomposition of a continuous linear map with a linear isometry preserves
the operator norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_comp_linear_isometry_equiv (f : F →SL[σ₂₃] G) (g : F' ≃ₛₗᵢ[σ₂'] F) :
‖f.comp g.to_linear_isometry.to_continuous_linear_map‖ = ‖f‖ | begin
casesI subsingleton_or_nontrivial F',
{ haveI := g.symm.to_linear_equiv.to_equiv.subsingleton,
simp },
refine le_antisymm _ _,
{ convert f.op_norm_comp_le g.to_linear_isometry.to_continuous_linear_map,
simp [g.to_linear_isometry.norm_to_continuous_linear_map] },
{ convert (f.comp g.to_linear_iso... | lemma | continuous_linear_map.op_norm_comp_linear_isometry_equiv | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"subsingleton_or_nontrivial"
] | Precomposition with a linear isometry preserves the operator norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_smul_right_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) :
‖smul_right c f‖ = ‖c‖ * ‖f‖ | begin
refine le_antisymm _ _,
{ apply op_norm_le_bound _ (mul_nonneg (norm_nonneg _) (norm_nonneg _)) (λx, _),
calc
‖(c x) • f‖ = ‖c x‖ * ‖f‖ : norm_smul _ _
... ≤ (‖c‖ * ‖x‖) * ‖f‖ :
mul_le_mul_of_nonneg_right (le_op_norm _ _) (norm_nonneg _)
... = ‖c‖ * ‖f‖ * ‖x‖ : by ring },
{ by_case... | lemma | continuous_linear_map.norm_smul_right_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"div_mul_eq_mul_div",
"div_nonneg",
"le_div_iff",
"mul_le_mul_of_nonneg_right",
"norm_smul",
"ring"
] | The norm of the tensor product of a scalar linear map and of an element of a normed space
is the product of the norms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_smul_right_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) :
‖smul_right c f‖₊ = ‖c‖₊ * ‖f‖₊ | nnreal.eq $ c.norm_smul_right_apply f | lemma | continuous_linear_map.nnnorm_smul_right_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"nnreal.eq"
] | The non-negative norm of the tensor product of a scalar linear map and of an element of a normed
space is the product of the non-negative norms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_rightL : (E →L[𝕜] 𝕜) →L[𝕜] Fₗ →L[𝕜] E →L[𝕜] Fₗ | linear_map.mk_continuous₂
{ to_fun := smul_rightₗ,
map_add' := λ c₁ c₂, by { ext x, simp only [add_smul, coe_smul_rightₗ, add_apply,
smul_right_apply, linear_map.add_apply] },
map_smul' := λ m c, by { ext x, simp only [smul_smul, coe_smul_rightₗ, algebra.id.smul_... | def | continuous_linear_map.smul_rightL | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"add_smul",
"algebra.id.smul_eq_mul",
"linear_map.add_apply",
"linear_map.coe_mk",
"linear_map.mk_continuous₂",
"linear_map.smul_apply",
"one_mul",
"pi.smul_apply",
"ring_hom.id_apply",
"smul_smul"
] | `continuous_linear_map.smul_right` as a continuous trilinear map:
`smul_rightL (c : E →L[𝕜] 𝕜) (f : F) (x : E) = c x • f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_smul_rightL_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) :
‖smul_rightL 𝕜 E Fₗ c f‖ = ‖c‖ * ‖f‖ | norm_smul_right_apply c f | lemma | continuous_linear_map.norm_smul_rightL_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_smul_rightL (c : E →L[𝕜] 𝕜) [nontrivial Fₗ] :
‖smul_rightL 𝕜 E Fₗ c‖ = ‖c‖ | continuous_linear_map.homothety_norm _ c.norm_smul_right_apply | lemma | continuous_linear_map.norm_smul_rightL | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.homothety_norm",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_mul [norm_one_class 𝕜'] : ‖mul 𝕜 𝕜'‖ = 1 | by haveI := norm_one_class.nontrivial 𝕜'; exact (mulₗᵢ 𝕜 𝕜').norm_to_continuous_linear_map | lemma | continuous_linear_map.op_norm_mul | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"norm_one_class",
"norm_one_class.nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_lsmul [normed_field 𝕜'] [normed_algebra 𝕜 𝕜']
[normed_space 𝕜' E] [is_scalar_tower 𝕜 𝕜' E] [nontrivial E] :
‖(lsmul 𝕜 𝕜' : 𝕜' →L[𝕜] E →L[𝕜] E)‖ = 1 | begin
refine continuous_linear_map.op_norm_eq_of_bounds zero_le_one (λ x, _) (λ N hN h, _),
{ rw one_mul,
exact op_norm_lsmul_apply_le _, },
obtain ⟨y, hy⟩ := exists_ne (0 : E),
have := le_of_op_norm_le _ (h 1) y,
simp_rw [lsmul_apply, one_smul, norm_one, mul_one] at this,
refine le_of_mul_le_mul_right ... | lemma | continuous_linear_map.op_norm_lsmul | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.op_norm_eq_of_bounds",
"exists_ne",
"is_scalar_tower",
"le_of_mul_le_mul_right",
"mul_one",
"nontrivial",
"normed_algebra",
"normed_field",
"normed_space",
"one_mul",
"one_smul",
"zero_le_one"
] | The norm of `lsmul` equals 1 in any nontrivial normed group.
This is `continuous_linear_map.op_norm_lsmul_le` as an equality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_subtypeL (K : submodule 𝕜 E) [nontrivial K] : ‖K.subtypeL‖ = 1 | K.subtypeₗᵢ.norm_to_continuous_linear_map | lemma | submodule.norm_subtypeL | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"nontrivial",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz (e : E ≃SL[σ₁₂] F) :
antilipschitz_with ‖(e.symm : F →SL[σ₂₁] E)‖₊ e | e.symm.lipschitz.to_right_inverse e.left_inv | lemma | continuous_linear_equiv.antilipschitz | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"antilipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_norm_mul_norm_symm [ring_hom_isometric σ₁₂] [nontrivial E] (e : E ≃SL[σ₁₂] F) :
1 ≤ ‖(e : E →SL[σ₁₂] F)‖ * ‖(e.symm : F →SL[σ₂₁] E)‖ | begin
rw [mul_comm],
convert (e.symm : F →SL[σ₂₁] E).op_norm_comp_le (e : E →SL[σ₁₂] F),
rw [e.coe_symm_comp_coe, continuous_linear_map.norm_id]
end | lemma | continuous_linear_equiv.one_le_norm_mul_norm_symm | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.norm_id",
"mul_comm",
"nontrivial",
"ring_hom_isometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_pos [ring_hom_isometric σ₁₂] [nontrivial E] (e : E ≃SL[σ₁₂] F) :
0 < ‖(e : E →SL[σ₁₂] F)‖ | pos_of_mul_pos_left (lt_of_lt_of_le zero_lt_one e.one_le_norm_mul_norm_symm) (norm_nonneg _) | lemma | continuous_linear_equiv.norm_pos | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"nontrivial",
"pos_of_mul_pos_left",
"ring_hom_isometric",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_symm_pos [ring_hom_isometric σ₁₂] [nontrivial E] (e : E ≃SL[σ₁₂] F) :
0 < ‖(e.symm : F →SL[σ₂₁] E)‖ | pos_of_mul_pos_right (zero_lt_one.trans_le e.one_le_norm_mul_norm_symm) (norm_nonneg _) | lemma | continuous_linear_equiv.norm_symm_pos | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"nontrivial",
"pos_of_mul_pos_right",
"ring_hom_isometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_symm_pos [ring_hom_isometric σ₁₂] [nontrivial E] (e : E ≃SL[σ₁₂] F) :
0 < ‖(e.symm : F →SL[σ₂₁] E)‖₊ | e.norm_symm_pos | lemma | continuous_linear_equiv.nnnorm_symm_pos | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"nontrivial",
"ring_hom_isometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_or_norm_symm_pos [ring_hom_isometric σ₁₂] (e : E ≃SL[σ₁₂] F) :
subsingleton E ∨ 0 < ‖(e.symm : F →SL[σ₂₁] E)‖ | begin
rcases subsingleton_or_nontrivial E with _i|_i; resetI,
{ left, apply_instance },
{ right, exact e.norm_symm_pos }
end | lemma | continuous_linear_equiv.subsingleton_or_norm_symm_pos | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"ring_hom_isometric",
"subsingleton_or_nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_or_nnnorm_symm_pos [ring_hom_isometric σ₁₂] (e : E ≃SL[σ₁₂] F) :
subsingleton E ∨ 0 < ‖(e.symm : F →SL[σ₂₁] E)‖₊ | subsingleton_or_norm_symm_pos e | lemma | continuous_linear_equiv.subsingleton_or_nnnorm_symm_pos | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"ring_hom_isometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_norm (x : E) (h : x ≠ 0) : ‖coord 𝕜 x h‖ = ‖x‖⁻¹ | begin
have hx : 0 < ‖x‖ := (norm_pos_iff.mpr h),
haveI : nontrivial (𝕜 ∙ x) := submodule.nontrivial_span_singleton h,
exact continuous_linear_map.homothety_norm _
(λ y, homothety_inverse _ hx _ (to_span_nonzero_singleton_homothety 𝕜 x h) _)
end | lemma | continuous_linear_equiv.coord_norm | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.homothety_norm",
"nontrivial",
"submodule.nontrivial_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coercive
[normed_add_comm_group E] [normed_space ℝ E]
(B : E →L[ℝ] E →L[ℝ] ℝ) : Prop | ∃ C, (0 < C) ∧ ∀ u, C * ‖u‖ * ‖u‖ ≤ B u u | def | is_coercive | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"normed_add_comm_group",
"normed_space"
] | A bounded bilinear form `B` in a real normed space is *coercive*
if there is some positive constant C such that `C * ‖u‖ * ‖u‖ ≤ B u u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_Lp (p : ℝ≥0∞) {ι : Type*} (α : ι → Type*) : Type* | Π (i : ι), α i | def | pi_Lp | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [] | A copy of a Pi type, on which we will put the `L^p` distance. Since the Pi type itself is
already endowed with the `L^∞` distance, we need the type synonym to avoid confusing typeclass
resolution. Also, we let it depend on `p`, to get a whole family of type on which we can put
different distances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv : pi_Lp p α ≃ Π (i : ι), α i | equiv.refl _ | def | pi_Lp.equiv | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"equiv",
"equiv.refl",
"pi_Lp"
] | Canonical bijection between `pi_Lp p α` and the original Pi type. We introduce it to be able
to compare the `L^p` and `L^∞` distances through it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_apply (x : pi_Lp p α) (i : ι) : pi_Lp.equiv p α x i = x i | rfl | lemma | pi_Lp.equiv_apply | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp",
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_symm_apply (x : Π i, α i) (i : ι) : (pi_Lp.equiv p α).symm x i = x i | rfl | lemma | pi_Lp.equiv_symm_apply | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp.equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_eq_card (f g : pi_Lp 0 β) : edist f g = {i | f i ≠ g i}.to_finite.to_finset.card | if_pos rfl | lemma | pi_Lp.edist_eq_card | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_eq_sum {p : ℝ≥0∞} (hp : 0 < p.to_real) (f g : pi_Lp p β) :
edist f g = (∑ i, edist (f i) (g i) ^ p.to_real) ^ (1/p.to_real) | let hp' := ennreal.to_real_pos_iff.mp hp in (if_neg hp'.1.ne').trans (if_neg hp'.2.ne) | lemma | pi_Lp.edist_eq_sum | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_eq_supr (f g : pi_Lp ∞ β) : edist f g = ⨆ i, edist (f i) (g i) | by { dsimp [edist], exact if_neg ennreal.top_ne_zero } | lemma | pi_Lp.edist_eq_supr | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"ennreal.top_ne_zero",
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_self (f : pi_Lp p β) : edist f f = 0 | begin
rcases p.trichotomy with (rfl | rfl | h),
{ simp [edist_eq_card], },
{ simp [edist_eq_supr], },
{ simp [edist_eq_sum h, ennreal.zero_rpow_of_pos h, ennreal.zero_rpow_of_pos (inv_pos.2 $ h)]}
end | lemma | pi_Lp.edist_self | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"ennreal.zero_rpow_of_pos",
"pi_Lp"
] | This holds independent of `p` and does not require `[fact (1 ≤ p)]`. We keep it separate
from `pi_Lp.pseudo_emetric_space` so it can be used also for `p < 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_comm (f g : pi_Lp p β) : edist f g = edist g f | begin
rcases p.trichotomy with (rfl | rfl | h),
{ simp only [edist_eq_card, eq_comm, ne.def] },
{ simp only [edist_eq_supr, edist_comm] },
{ simp only [edist_eq_sum h, edist_comm] }
end | lemma | pi_Lp.edist_comm | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp"
] | This holds independent of `p` and does not require `[fact (1 ≤ p)]`. We keep it separate
from `pi_Lp.pseudo_emetric_space` so it can be used also for `p < 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_eq_card (f g : pi_Lp 0 α) : dist f g = {i | f i ≠ g i}.to_finite.to_finset.card | if_pos rfl | lemma | pi_Lp.dist_eq_card | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq_sum {p : ℝ≥0∞} (hp : 0 < p.to_real) (f g : pi_Lp p α) :
dist f g = (∑ i, dist (f i) (g i) ^ p.to_real) ^ (1/p.to_real) | let hp' := ennreal.to_real_pos_iff.mp hp in (if_neg hp'.1.ne').trans (if_neg hp'.2.ne) | lemma | pi_Lp.dist_eq_sum | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq_csupr (f g : pi_Lp ∞ α) : dist f g = ⨆ i, dist (f i) (g i) | by { dsimp [dist], exact if_neg ennreal.top_ne_zero } | lemma | pi_Lp.dist_eq_csupr | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"ennreal.top_ne_zero",
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_norm : has_norm (pi_Lp p β) | { norm := λ f, if hp : p = 0 then {i | f i ≠ 0}.to_finite.to_finset.card
else (if p = ∞ then ⨆ i, ‖f i‖ else (∑ i, ‖f i‖ ^ p.to_real) ^ (1 / p.to_real)) } | instance | pi_Lp.has_norm | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"has_norm",
"pi_Lp"
] | Endowing the space `pi_Lp p β` with the `L^p` norm. We register this instance
separate from `pi_Lp.seminormed_add_comm_group` since the latter requires the type class hypothesis
`[fact (1 ≤ p)]` in order to prove the triangle inequality.
Registering this separately allows for a future norm-like structure on `pi_Lp p β... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_eq_card (f : pi_Lp 0 β) : ‖f‖ = {i | f i ≠ 0}.to_finite.to_finset.card | if_pos rfl | lemma | pi_Lp.norm_eq_card | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_csupr (f : pi_Lp ∞ β) : ‖f‖ = ⨆ i, ‖f i‖ | by { dsimp [norm], exact if_neg ennreal.top_ne_zero } | lemma | pi_Lp.norm_eq_csupr | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"ennreal.top_ne_zero",
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_eq_sum (hp : 0 < p.to_real) (f : pi_Lp p β) :
‖f‖ = (∑ i, ‖f i‖ ^ p.to_real) ^ (1 / p.to_real) | let hp' := ennreal.to_real_pos_iff.mp hp in (if_neg hp'.1.ne').trans (if_neg hp'.2.ne) | lemma | pi_Lp.norm_eq_sum | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_emetric_aux : pseudo_emetric_space (pi_Lp p β) | { edist_self := pi_Lp.edist_self p,
edist_comm := pi_Lp.edist_comm p,
edist_triangle := λ f g h,
begin
unfreezingI { rcases p.dichotomy with (rfl | hp) },
{ simp only [edist_eq_supr],
casesI is_empty_or_nonempty ι,
{ simp only [csupr_of_empty, ennreal.bot_eq_zero, add_zero, nonpos_iff_eq_zero]... | def | pi_Lp.pseudo_emetric_aux | analysis.normed_space | src/analysis/normed_space/pi_Lp.lean | [
"analysis.mean_inequalities",
"data.fintype.order",
"linear_algebra.matrix.basis"
] | [
"csupr_of_empty",
"ennreal.Lp_add_le",
"ennreal.bot_eq_zero",
"ennreal.rpow_le_rpow",
"is_empty_or_nonempty",
"le_supr",
"pi_Lp",
"pi_Lp.edist_comm",
"pi_Lp.edist_self",
"pseudo_emetric_space",
"supr_le"
] | Endowing the space `pi_Lp p β` with the `L^p` pseudoemetric structure. This definition is not
satisfactory, as it does not register the fact that the topology and the uniform structure coincide
with the product one. Therefore, we do not register it as an instance. Using this as a temporary
pseudoemetric space instance,... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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