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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is_O_sub {f : E → F} (h : is_bounded_linear_map 𝕜 f)
(l : filter E) (x : E) : (λ x', f (x' - x)) =O[l] (λ x', x' - x) | is_O_comp h l | theorem | is_bounded_linear_map.is_O_sub | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_linear_map_prod_multilinear
{E : ι → Type*} [∀ i, normed_add_comm_group (E i)] [∀ i, normed_space 𝕜 (E i)] :
is_bounded_linear_map 𝕜
(λ p : (continuous_multilinear_map 𝕜 E F) × (continuous_multilinear_map 𝕜 E G), p.1.prod p.2) | { map_add := λ p₁ p₂, by { ext1 m, refl },
map_smul := λ c p, by { ext1 m, refl },
bound := ⟨1, zero_lt_one, λ p, begin
rw one_mul,
apply continuous_multilinear_map.op_norm_le_bound _ (norm_nonneg _) (λ m, _),
rw [continuous_multilinear_map.prod_apply, norm_prod_le_iff],
split,
{ exact (p.1.le_o... | lemma | is_bounded_linear_map_prod_multilinear | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"bound",
"continuous_multilinear_map",
"continuous_multilinear_map.op_norm_le_bound",
"continuous_multilinear_map.prod_apply",
"finset.prod_nonneg",
"is_bounded_linear_map",
"mul_le_mul_of_nonneg_right",
"norm_fst_le",
"norm_prod_le_iff",
"norm_snd_le",
"normed_add_comm_group",
"normed_space",... | Taking the cartesian product of two continuous multilinear maps
is a bounded linear operation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_linear_map_continuous_multilinear_map_comp_linear (g : G →L[𝕜] E) :
is_bounded_linear_map 𝕜 (λ f : continuous_multilinear_map 𝕜 (λ (i : ι), E) F,
f.comp_continuous_linear_map (λ _, g)) | begin
refine is_linear_map.with_bound ⟨λ f₁ f₂, by { ext m, refl }, λ c f, by { ext m, refl }⟩
(‖g‖ ^ (fintype.card ι)) (λ f, _),
apply continuous_multilinear_map.op_norm_le_bound _ _ (λ m, _),
{ apply_rules [mul_nonneg, pow_nonneg, norm_nonneg] },
calc ‖f (g ∘ m)‖ ≤
‖f‖ * ∏ i, ‖g (m i)‖ : f.le_op_norm ... | lemma | is_bounded_linear_map_continuous_multilinear_map_comp_linear | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous_multilinear_map",
"continuous_multilinear_map.op_norm_le_bound",
"finset.card_univ",
"finset.prod_le_prod",
"finset.prod_mul_distrib",
"fintype.card",
"is_bounded_linear_map",
"is_linear_map.with_bound",
"mul_le_mul_of_nonneg_left",
"pow_nonneg",
"ring"
] | Given a fixed continuous linear map `g`, associating to a continuous multilinear map `f` the
continuous multilinear map `f (g m₁, ..., g mₙ)` is a bounded linear operation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_add₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (x x' : M) (y : F) :
f (x + x') y = f x y + f x' y | by rw [f.map_add, add_apply] | lemma | continuous_linear_map.map_add₂ | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (y : F) : f 0 y = 0 | by rw [f.map_zero, zero_apply] | lemma | continuous_linear_map.map_zero₂ | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smulₛₗ₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (c : R) (x : M) (y : F) :
f (c • x) y = ρ₁₂ c • f x y | by rw [f.map_smulₛₗ, smul_apply] | lemma | continuous_linear_map.map_smulₛₗ₂ | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (x x' : M) (y : F) :
f (x - x') y = f x y - f x' y | by rw [f.map_sub, sub_apply] | lemma | continuous_linear_map.map_sub₂ | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (x : M) (y : F) : f (- x) y = - f x y | by rw [f.map_neg, neg_apply] | lemma | continuous_linear_map.map_neg₂ | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul₂ (f : E →L[𝕜] F →L[𝕜] G) (c : 𝕜) (x : E) (y : F) : f (c • x) y = c • f x y | by rw [f.map_smul, smul_apply] | lemma | continuous_linear_map.map_smul₂ | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map (f : E × F → G) : Prop | (add_left : ∀ (x₁ x₂ : E) (y : F), f (x₁ + x₂, y) = f (x₁, y) + f (x₂, y))
(smul_left : ∀ (c : 𝕜) (x : E) (y : F), f (c • x, y) = c • f (x, y))
(add_right : ∀ (x : E) (y₁ y₂ : F), f (x, y₁ + y₂) = f (x, y₁) + f (x, y₂))
(smul_right : ∀ (c : 𝕜) (x : E) (y : F), f (x, c • y) = c • f (x,y))
(bound : ∃ C > 0, ∀ ... | structure | is_bounded_bilinear_map | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"bound"
] | A map `f : E × F → G` satisfies `is_bounded_bilinear_map 𝕜 f` if it is bilinear and
continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.is_bounded_bilinear_map (f : E →L[𝕜] F →L[𝕜] G) :
is_bounded_bilinear_map 𝕜 (λ x : E × F, f x.1 x.2) | { add_left := f.map_add₂,
smul_left := f.map_smul₂,
add_right := λ x, (f x).map_add,
smul_right := λ c x, (f x).map_smul c,
bound := ⟨max ‖f‖ 1, zero_lt_one.trans_le (le_max_right _ _),
λ x y, (f.le_op_norm₂ x y).trans $
by apply_rules [mul_le_mul_of_nonneg_right, norm_nonneg, le_max_left]⟩ } | lemma | continuous_linear_map.is_bounded_bilinear_map | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"bound",
"is_bounded_bilinear_map",
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.is_O (h : is_bounded_bilinear_map 𝕜 f) :
f =O[⊤] (λ p : E × F, ‖p.1‖ * ‖p.2‖) | let ⟨C, Cpos, hC⟩ := h.bound in asymptotics.is_O.of_bound _ $
filter.eventually_of_forall $ λ ⟨x, y⟩, by simpa [mul_assoc] using hC x y | lemma | is_bounded_bilinear_map.is_O | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"asymptotics.is_O.of_bound",
"filter.eventually_of_forall",
"is_bounded_bilinear_map",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.is_O_comp {α : Type*} (H : is_bounded_bilinear_map 𝕜 f)
{g : α → E} {h : α → F} {l : filter α} :
(λ x, f (g x, h x)) =O[l] (λ x, ‖g x‖ * ‖h x‖) | H.is_O.comp_tendsto le_top | lemma | is_bounded_bilinear_map.is_O_comp | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"is_bounded_bilinear_map",
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.is_O' (h : is_bounded_bilinear_map 𝕜 f) :
f =O[⊤] (λ p : E × F, ‖p‖ * ‖p‖) | h.is_O.trans $ (@asymptotics.is_O_fst_prod' _ E F _ _ _ _).norm_norm.mul
(@asymptotics.is_O_snd_prod' _ E F _ _ _ _).norm_norm | lemma | is_bounded_bilinear_map.is_O' | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"asymptotics.is_O_fst_prod'",
"asymptotics.is_O_snd_prod'",
"is_bounded_bilinear_map",
"norm_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.map_sub_left (h : is_bounded_bilinear_map 𝕜 f) {x y : E} {z : F} :
f (x - y, z) = f (x, z) - f(y, z) | calc f (x - y, z) = f (x + (-1 : 𝕜) • y, z) : by simp [sub_eq_add_neg]
... = f (x, z) + (-1 : 𝕜) • f (y, z) : by simp only [h.add_left, h.smul_left]
... = f (x, z) - f (y, z) : by simp [sub_eq_add_neg] | lemma | is_bounded_bilinear_map.map_sub_left | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.map_sub_right (h : is_bounded_bilinear_map 𝕜 f) {x : E} {y z : F} :
f (x, y - z) = f (x, y) - f (x, z) | calc f (x, y - z) = f (x, y + (-1 : 𝕜) • z) : by simp [sub_eq_add_neg]
... = f (x, y) + (-1 : 𝕜) • f (x, z) : by simp only [h.add_right, h.smul_right]
... = f (x, y) - f (x, z) : by simp [sub_eq_add_neg] | lemma | is_bounded_bilinear_map.map_sub_right | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.continuous (h : is_bounded_bilinear_map 𝕜 f) :
continuous f | begin
have one_ne : (1:ℝ) ≠ 0 := by simp,
obtain ⟨C, (Cpos : 0 < C), hC⟩ := h.bound,
rw continuous_iff_continuous_at,
intros x,
have H : ∀ (a:E) (b:F), ‖f (a, b)‖ ≤ C * ‖‖a‖ * ‖b‖‖,
{ intros a b,
simpa [mul_assoc] using hC a b },
have h₁ : (λ e : E × F, f (e.1 - x.1, e.2)) =o[𝓝 x] (λ e, (1:ℝ)),
{ r... | lemma | is_bounded_bilinear_map.continuous | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"asymptotics.is_O_of_le'",
"asymptotics.is_o_const_iff",
"continuous",
"continuous_at",
"continuous_const",
"continuous_iff_continuous_at",
"is_bounded_bilinear_map",
"mul_assoc"
] | Useful to use together with `continuous.comp₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_bilinear_map.continuous_left (h : is_bounded_bilinear_map 𝕜 f) {e₂ : F} :
continuous (λe₁, f (e₁, e₂)) | h.continuous.comp (continuous_id.prod_mk continuous_const) | lemma | is_bounded_bilinear_map.continuous_left | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous",
"continuous_const",
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.continuous_right (h : is_bounded_bilinear_map 𝕜 f) {e₁ : E} :
continuous (λe₂, f (e₁, e₂)) | h.continuous.comp (continuous_const.prod_mk continuous_id) | lemma | is_bounded_bilinear_map.continuous_right | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous",
"continuous_id",
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.continuous₂ (f : E →L[𝕜] F →L[𝕜] G) :
continuous (function.uncurry (λ x y, f x y)) | f.is_bounded_bilinear_map.continuous | lemma | continuous_linear_map.continuous₂ | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous"
] | Useful to use together with `continuous.comp₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_bilinear_map.is_bounded_linear_map_left (h : is_bounded_bilinear_map 𝕜 f) (y : F) :
is_bounded_linear_map 𝕜 (λ x, f (x, y)) | { map_add := λ x x', h.add_left _ _ _,
map_smul := λ c x, h.smul_left _ _ _,
bound := begin
rcases h.bound with ⟨C, C_pos, hC⟩,
refine ⟨C * (‖y‖ + 1), mul_pos C_pos (lt_of_lt_of_le (zero_lt_one) (by simp)), λ x, _⟩,
have : ‖y‖ ≤ ‖y‖ + 1, by simp [zero_le_one],
calc ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖ : h... | lemma | is_bounded_bilinear_map.is_bounded_linear_map_left | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"bound",
"is_bounded_bilinear_map",
"is_bounded_linear_map",
"mul_le_mul_of_nonneg_left",
"ring",
"zero_le_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.is_bounded_linear_map_right
(h : is_bounded_bilinear_map 𝕜 f) (x : E) :
is_bounded_linear_map 𝕜 (λ y, f (x, y)) | { map_add := λ y y', h.add_right _ _ _,
map_smul := λ c y, h.smul_right _ _ _,
bound := begin
rcases h.bound with ⟨C, C_pos, hC⟩,
refine ⟨C * (‖x‖ + 1), mul_pos C_pos (lt_of_lt_of_le (zero_lt_one) (by simp)), λ y, _⟩,
have : ‖x‖ ≤ ‖x‖ + 1, by simp [zero_le_one],
calc ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖ :... | lemma | is_bounded_bilinear_map.is_bounded_linear_map_right | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"bound",
"is_bounded_bilinear_map",
"is_bounded_linear_map",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"zero_le_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map_smul {𝕜' : Type*} [normed_field 𝕜']
[normed_algebra 𝕜 𝕜'] {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] [normed_space 𝕜' E]
[is_scalar_tower 𝕜 𝕜' E] :
is_bounded_bilinear_map 𝕜 (λ (p : 𝕜' × E), p.1 • p.2) | (lsmul 𝕜 𝕜' : 𝕜' →L[𝕜] E →L[𝕜] E).is_bounded_bilinear_map | lemma | is_bounded_bilinear_map_smul | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_bilinear_map",
"is_scalar_tower",
"normed_add_comm_group",
"normed_algebra",
"normed_field",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map_mul :
is_bounded_bilinear_map 𝕜 (λ (p : 𝕜 × 𝕜), p.1 * p.2) | by simp_rw ← smul_eq_mul; exact is_bounded_bilinear_map_smul | lemma | is_bounded_bilinear_map_mul | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_bilinear_map",
"is_bounded_bilinear_map_smul",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map_comp :
is_bounded_bilinear_map 𝕜 (λ (p : (F →L[𝕜] G) × (E →L[𝕜] F)), p.1.comp p.2) | (compL 𝕜 E F G).is_bounded_bilinear_map | lemma | is_bounded_bilinear_map_comp | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.is_bounded_linear_map_comp_left (g : F →L[𝕜] G) :
is_bounded_linear_map 𝕜 (λ (f : E →L[𝕜] F), continuous_linear_map.comp g f) | is_bounded_bilinear_map_comp.is_bounded_linear_map_right _ | lemma | continuous_linear_map.is_bounded_linear_map_comp_left | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous_linear_map.comp",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.is_bounded_linear_map_comp_right (f : E →L[𝕜] F) :
is_bounded_linear_map 𝕜 (λ (g : F →L[𝕜] G), continuous_linear_map.comp g f) | is_bounded_bilinear_map_comp.is_bounded_linear_map_left _ | lemma | continuous_linear_map.is_bounded_linear_map_comp_right | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous_linear_map.comp",
"is_bounded_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map_apply :
is_bounded_bilinear_map 𝕜 (λ p : (E →L[𝕜] F) × E, p.1 p.2) | (continuous_linear_map.flip (apply 𝕜 F : E →L[𝕜] (E →L[𝕜] F) →L[𝕜] F)).is_bounded_bilinear_map | lemma | is_bounded_bilinear_map_apply | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous_linear_map.flip",
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map_smul_right :
is_bounded_bilinear_map 𝕜
(λ p, (continuous_linear_map.smul_right : (E →L[𝕜] 𝕜) → F → (E →L[𝕜] F)) p.1 p.2) | (smul_rightL 𝕜 E F).is_bounded_bilinear_map | lemma | is_bounded_bilinear_map_smul_right | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous_linear_map.smul_right",
"is_bounded_bilinear_map"
] | The function `continuous_linear_map.smul_right`, associating to a continuous linear map
`f : E → 𝕜` and a scalar `c : F` the tensor product `f ⊗ c` as a continuous linear map from `E` to
`F`, is a bounded bilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_bilinear_map_comp_multilinear {ι : Type*} {E : ι → Type*}
[fintype ι] [∀ i, normed_add_comm_group (E i)] [∀ i, normed_space 𝕜 (E i)] :
is_bounded_bilinear_map 𝕜 (λ p : (F →L[𝕜] G) × (continuous_multilinear_map 𝕜 E F),
p.1.comp_continuous_multilinear_map p.2) | (comp_continuous_multilinear_mapL 𝕜 E F G).is_bounded_bilinear_map | lemma | is_bounded_bilinear_map_comp_multilinear | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous_multilinear_map",
"fintype",
"is_bounded_bilinear_map",
"normed_add_comm_group",
"normed_space"
] | The composition of a continuous linear map with a continuous multilinear map is a bounded
bilinear operation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_bilinear_map.linear_deriv (h : is_bounded_bilinear_map 𝕜 f) (p : E × F) :
E × F →ₗ[𝕜] G | { to_fun := λ q, f (p.1, q.2) + f (q.1, p.2),
map_add' := λ q₁ q₂, begin
change f (p.1, q₁.2 + q₂.2) + f (q₁.1 + q₂.1, p.2) =
f (p.1, q₁.2) + f (q₁.1, p.2) + (f (p.1, q₂.2) + f (q₂.1, p.2)),
simp [h.add_left, h.add_right], abel
end,
map_smul' := λ c q, begin
change f (p.1, c • q.2) + f (c • q.1,... | def | is_bounded_bilinear_map.linear_deriv | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_bilinear_map",
"smul_add"
] | Definition of the derivative of a bilinear map `f`, given at a point `p` by
`q ↦ f(p.1, q.2) + f(q.1, p.2)` as in the standard formula for the derivative of a product.
We define this function here as a linear map `E × F →ₗ[𝕜] G`, then `is_bounded_bilinear_map.deriv`
strengthens it to a continuous linear map `E × F →L[... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_bilinear_map.deriv (h : is_bounded_bilinear_map 𝕜 f) (p : E × F) : E × F →L[𝕜] G | (h.linear_deriv p).mk_continuous_of_exists_bound $ begin
rcases h.bound with ⟨C, Cpos, hC⟩,
refine ⟨C * ‖p.1‖ + C * ‖p.2‖, λ q, _⟩,
calc ‖f (p.1, q.2) + f (q.1, p.2)‖
≤ C * ‖p.1‖ * ‖q.2‖ + C * ‖q.1‖ * ‖p.2‖ : norm_add_le_of_le (hC _ _) (hC _ _)
... ≤ C * ‖p.1‖ * ‖q‖ + C * ‖q‖ * ‖p.2‖ : begin
apply add... | def | is_bounded_bilinear_map.deriv | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_bilinear_map",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"ring"
] | The derivative of a bounded bilinear map at a point `p : E × F`, as a continuous linear map
from `E × F` to `G`. The statement that this is indeed the derivative of `f` is
`is_bounded_bilinear_map.has_fderiv_at` in `analysis.calculus.fderiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_bilinear_map_deriv_coe (h : is_bounded_bilinear_map 𝕜 f) (p q : E × F) :
h.deriv p q = f (p.1, q.2) + f (q.1, p.2) | rfl | lemma | is_bounded_bilinear_map_deriv_coe | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.mul_left_right_is_bounded_bilinear
(𝕜' : Type*) [normed_ring 𝕜'] [normed_algebra 𝕜 𝕜'] :
is_bounded_bilinear_map 𝕜 (λ p : 𝕜' × 𝕜', continuous_linear_map.mul_left_right 𝕜 𝕜' p.1 p.2) | (continuous_linear_map.mul_left_right 𝕜 𝕜').is_bounded_bilinear_map | lemma | continuous_linear_map.mul_left_right_is_bounded_bilinear | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous_linear_map.mul_left_right",
"is_bounded_bilinear_map",
"normed_algebra",
"normed_ring"
] | The function `continuous_linear_map.mul_left_right : 𝕜' × 𝕜' → (𝕜' →L[𝕜] 𝕜')` is a bounded
bilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded_bilinear_map.is_bounded_linear_map_deriv (h : is_bounded_bilinear_map 𝕜 f) :
is_bounded_linear_map 𝕜 (λ p : E × F, h.deriv p) | begin
rcases h.bound with ⟨C, Cpos : 0 < C, hC⟩,
refine is_linear_map.with_bound ⟨λ p₁ p₂, _, λ c p, _⟩ (C + C) (λ p, _),
{ ext; simp only [h.add_left, h.add_right, coe_comp', function.comp_app, inl_apply,
is_bounded_bilinear_map_deriv_coe, prod.fst_add, prod.snd_add, add_apply]; abel },
{ ext; simp only [h... | lemma | is_bounded_bilinear_map.is_bounded_linear_map_deriv | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous_linear_map.op_norm_le_bound",
"is_bounded_bilinear_map",
"is_bounded_bilinear_map_deriv_coe",
"is_bounded_linear_map",
"is_linear_map.with_bound",
"mul_le_mul",
"pi.smul_apply",
"prod.smul_fst",
"prod.smul_snd",
"ring",
"smul_add"
] | Given a bounded bilinear map `f`, the map associating to a point `p` the derivative of `f` at
`p` is itself a bounded linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.clm_comp {X} [topological_space X] {g : X → F →L[𝕜] G} {f : X → E →L[𝕜] F}
(hg : continuous g) (hf : continuous f) :
continuous (λ x, (g x).comp (f x)) | (compL 𝕜 E F G).continuous₂.comp₂ hg hf | lemma | continuous.clm_comp | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.clm_comp {X} [topological_space X] {g : X → F →L[𝕜] G} {f : X → E →L[𝕜] F}
{s : set X} (hg : continuous_on g s) (hf : continuous_on f s) :
continuous_on (λ x, (g x).comp (f x)) s | (compL 𝕜 E F G).continuous₂.comp_continuous_on (hg.prod hf) | lemma | continuous_on.clm_comp | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"continuous_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open [complete_space E] : is_open (range (coe : (E ≃L[𝕜] F) → (E →L[𝕜] F))) | begin
rw [is_open_iff_mem_nhds, forall_range_iff],
refine λ e, is_open.mem_nhds _ (mem_range_self _),
let O : (E →L[𝕜] F) → (E →L[𝕜] E) := λ f, (e.symm : F →L[𝕜] E).comp f,
have h_O : continuous O := is_bounded_bilinear_map_comp.continuous_right,
convert show is_open (O ⁻¹' {x | is_unit x}), from units.is_... | lemma | continuous_linear_equiv.is_open | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"coe_fn_coe_base'",
"complete_space",
"continuous",
"is_open",
"is_open.mem_nhds",
"is_open_iff_mem_nhds",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds [complete_space E] (e : E ≃L[𝕜] F) :
(range (coe : (E ≃L[𝕜] F) → (E →L[𝕜] F))) ∈ 𝓝 (e : E →L[𝕜] F) | is_open.mem_nhds continuous_linear_equiv.is_open (by simp) | lemma | continuous_linear_equiv.nhds | analysis.normed_space | src/analysis/normed_space/bounded_linear_maps.lean | [
"analysis.normed_space.multilinear",
"analysis.normed_space.units",
"analysis.asymptotics.asymptotics"
] | [
"complete_space",
"continuous_linear_equiv.is_open",
"is_open.mem_nhds",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator {M₁ M₂ : Type*} [has_zero M₁] [topological_space M₁]
[topological_space M₂] (f : M₁ → M₂) : Prop | ∃ K, is_compact K ∧ f ⁻¹' K ∈ (𝓝 0 : filter M₁) | def | is_compact_operator | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"filter",
"is_compact",
"topological_space"
] | A compact operator between two topological vector spaces. This definition is usually
given as "there exists a neighborhood of zero whose image is contained in a compact set",
but we choose a definition which involves fewer existential quantifiers and replaces images
with preimages.
We prove the equivalence in `is_comp... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_operator_zero {M₁ M₂ : Type*} [has_zero M₁] [topological_space M₁]
[topological_space M₂] [has_zero M₂] : is_compact_operator (0 : M₁ → M₂) | ⟨{0}, is_compact_singleton, mem_of_superset univ_mem (λ x _, rfl)⟩ | lemma | is_compact_operator_zero | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"is_compact_operator",
"is_compact_singleton",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator_iff_exists_mem_nhds_image_subset_compact (f : M₁ → M₂) :
is_compact_operator f ↔ ∃ V ∈ (𝓝 0 : filter M₁), ∃ (K : set M₂), is_compact K ∧ f '' V ⊆ K | ⟨λ ⟨K, hK, hKf⟩, ⟨f ⁻¹' K, hKf, K, hK, image_preimage_subset _ _⟩,
λ ⟨V, hV, K, hK, hVK⟩, ⟨K, hK, mem_of_superset hV (image_subset_iff.mp hVK)⟩⟩ | lemma | is_compact_operator_iff_exists_mem_nhds_image_subset_compact | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"filter",
"is_compact",
"is_compact_operator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator_iff_exists_mem_nhds_is_compact_closure_image [t2_space M₂]
(f : M₁ → M₂) :
is_compact_operator f ↔ ∃ V ∈ (𝓝 0 : filter M₁), is_compact (closure $ f '' V) | begin
rw is_compact_operator_iff_exists_mem_nhds_image_subset_compact,
exact ⟨λ ⟨V, hV, K, hK, hKV⟩, ⟨V, hV, is_compact_closure_of_subset_compact hK hKV⟩,
λ ⟨V, hV, hVc⟩, ⟨V, hV, closure (f '' V), hVc, subset_closure⟩⟩,
end | lemma | is_compact_operator_iff_exists_mem_nhds_is_compact_closure_image | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"closure",
"filter",
"is_compact",
"is_compact_closure_of_subset_compact",
"is_compact_operator",
"is_compact_operator_iff_exists_mem_nhds_image_subset_compact",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.image_subset_compact_of_vonN_bounded {f : M₁ →ₛₗ[σ₁₂] M₂}
(hf : is_compact_operator f) {S : set M₁} (hS : is_vonN_bounded 𝕜₁ S) :
∃ (K : set M₂), is_compact K ∧ f '' S ⊆ K | let ⟨K, hK, hKf⟩ := hf,
⟨r, hr, hrS⟩ := hS hKf,
⟨c, hc⟩ := normed_field.exists_lt_norm 𝕜₁ r,
this := ne_zero_of_norm_ne_zero (hr.trans hc).ne.symm in
⟨σ₁₂ c • K, hK.image $ continuous_id.const_smul (σ₁₂ c),
by rw [image_subset_iff, preimage_smul_setₛₗ _ _ _ f this.is_unit]; exact hrS c hc.le⟩ | lemma | is_compact_operator.image_subset_compact_of_vonN_bounded | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"is_compact",
"is_compact_operator",
"normed_field.exists_lt_norm",
"preimage_smul_setₛₗ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.is_compact_closure_image_of_vonN_bounded [t2_space M₂]
{f : M₁ →ₛₗ[σ₁₂] M₂} (hf : is_compact_operator f) {S : set M₁}
(hS : is_vonN_bounded 𝕜₁ S) : is_compact (closure $ f '' S) | let ⟨K, hK, hKf⟩ := hf.image_subset_compact_of_vonN_bounded hS in
is_compact_closure_of_subset_compact hK hKf | lemma | is_compact_operator.is_compact_closure_image_of_vonN_bounded | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"closure",
"is_compact",
"is_compact_closure_of_subset_compact",
"is_compact_operator",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.image_subset_compact_of_bounded [has_continuous_const_smul 𝕜₂ M₂]
{f : M₁ →ₛₗ[σ₁₂] M₂} (hf : is_compact_operator f) {S : set M₁} (hS : metric.bounded S) :
∃ (K : set M₂), is_compact K ∧ f '' S ⊆ K | hf.image_subset_compact_of_vonN_bounded
(by rwa [normed_space.is_vonN_bounded_iff, ← metric.bounded_iff_is_bounded]) | lemma | is_compact_operator.image_subset_compact_of_bounded | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"has_continuous_const_smul",
"is_compact",
"is_compact_operator",
"metric.bounded",
"metric.bounded_iff_is_bounded",
"normed_space.is_vonN_bounded_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.is_compact_closure_image_of_bounded [has_continuous_const_smul 𝕜₂ M₂]
[t2_space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : is_compact_operator f) {S : set M₁}
(hS : metric.bounded S) : is_compact (closure $ f '' S) | hf.is_compact_closure_image_of_vonN_bounded
(by rwa [normed_space.is_vonN_bounded_iff, ← metric.bounded_iff_is_bounded]) | lemma | is_compact_operator.is_compact_closure_image_of_bounded | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"closure",
"has_continuous_const_smul",
"is_compact",
"is_compact_operator",
"metric.bounded",
"metric.bounded_iff_is_bounded",
"normed_space.is_vonN_bounded_iff",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.image_ball_subset_compact [has_continuous_const_smul 𝕜₂ M₂]
{f : M₁ →ₛₗ[σ₁₂] M₂} (hf : is_compact_operator f) (r : ℝ) :
∃ (K : set M₂), is_compact K ∧ f '' metric.ball 0 r ⊆ K | hf.image_subset_compact_of_vonN_bounded (normed_space.is_vonN_bounded_ball 𝕜₁ M₁ r) | lemma | is_compact_operator.image_ball_subset_compact | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"has_continuous_const_smul",
"is_compact",
"is_compact_operator",
"metric.ball",
"normed_space.is_vonN_bounded_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.image_closed_ball_subset_compact [has_continuous_const_smul 𝕜₂ M₂]
{f : M₁ →ₛₗ[σ₁₂] M₂} (hf : is_compact_operator f) (r : ℝ) :
∃ (K : set M₂), is_compact K ∧ f '' metric.closed_ball 0 r ⊆ K | hf.image_subset_compact_of_vonN_bounded (normed_space.is_vonN_bounded_closed_ball 𝕜₁ M₁ r) | lemma | is_compact_operator.image_closed_ball_subset_compact | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"has_continuous_const_smul",
"is_compact",
"is_compact_operator",
"metric.closed_ball",
"normed_space.is_vonN_bounded_closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.is_compact_closure_image_ball [has_continuous_const_smul 𝕜₂ M₂]
[t2_space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : is_compact_operator f) (r : ℝ) :
is_compact (closure $ f '' metric.ball 0 r) | hf.is_compact_closure_image_of_vonN_bounded (normed_space.is_vonN_bounded_ball 𝕜₁ M₁ r) | lemma | is_compact_operator.is_compact_closure_image_ball | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"closure",
"has_continuous_const_smul",
"is_compact",
"is_compact_operator",
"metric.ball",
"normed_space.is_vonN_bounded_ball",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.is_compact_closure_image_closed_ball [has_continuous_const_smul 𝕜₂ M₂]
[t2_space M₂] {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : is_compact_operator f) (r : ℝ) :
is_compact (closure $ f '' metric.closed_ball 0 r) | hf.is_compact_closure_image_of_vonN_bounded (normed_space.is_vonN_bounded_closed_ball 𝕜₁ M₁ r) | lemma | is_compact_operator.is_compact_closure_image_closed_ball | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"closure",
"has_continuous_const_smul",
"is_compact",
"is_compact_operator",
"metric.closed_ball",
"normed_space.is_vonN_bounded_closed_ball",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator_iff_image_ball_subset_compact [has_continuous_const_smul 𝕜₂ M₂]
(f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r) :
is_compact_operator f ↔ ∃ (K : set M₂), is_compact K ∧ f '' metric.ball 0 r ⊆ K | ⟨λ hf, hf.image_ball_subset_compact r,
λ ⟨K, hK, hKr⟩, (is_compact_operator_iff_exists_mem_nhds_image_subset_compact f).mpr
⟨metric.ball 0 r, ball_mem_nhds _ hr, K, hK, hKr⟩⟩ | lemma | is_compact_operator_iff_image_ball_subset_compact | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"has_continuous_const_smul",
"is_compact",
"is_compact_operator",
"is_compact_operator_iff_exists_mem_nhds_image_subset_compact",
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator_iff_image_closed_ball_subset_compact [has_continuous_const_smul 𝕜₂ M₂]
(f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r) :
is_compact_operator f ↔ ∃ (K : set M₂), is_compact K ∧ f '' metric.closed_ball 0 r ⊆ K | ⟨λ hf, hf.image_closed_ball_subset_compact r,
λ ⟨K, hK, hKr⟩, (is_compact_operator_iff_exists_mem_nhds_image_subset_compact f).mpr
⟨metric.closed_ball 0 r, closed_ball_mem_nhds _ hr, K, hK, hKr⟩⟩ | lemma | is_compact_operator_iff_image_closed_ball_subset_compact | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"has_continuous_const_smul",
"is_compact",
"is_compact_operator",
"is_compact_operator_iff_exists_mem_nhds_image_subset_compact",
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator_iff_is_compact_closure_image_ball [has_continuous_const_smul 𝕜₂ M₂]
[t2_space M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r) :
is_compact_operator f ↔ is_compact (closure $ f '' metric.ball 0 r) | ⟨λ hf, hf.is_compact_closure_image_ball r,
λ hf, (is_compact_operator_iff_exists_mem_nhds_is_compact_closure_image f).mpr
⟨metric.ball 0 r, ball_mem_nhds _ hr, hf⟩⟩ | lemma | is_compact_operator_iff_is_compact_closure_image_ball | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"closure",
"has_continuous_const_smul",
"is_compact",
"is_compact_operator",
"is_compact_operator_iff_exists_mem_nhds_is_compact_closure_image",
"metric.ball",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator_iff_is_compact_closure_image_closed_ball
[has_continuous_const_smul 𝕜₂ M₂] [t2_space M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) {r : ℝ} (hr : 0 < r) :
is_compact_operator f ↔ is_compact (closure $ f '' metric.closed_ball 0 r) | ⟨λ hf, hf.is_compact_closure_image_closed_ball r,
λ hf, (is_compact_operator_iff_exists_mem_nhds_is_compact_closure_image f).mpr
⟨metric.closed_ball 0 r, closed_ball_mem_nhds _ hr, hf⟩⟩ | lemma | is_compact_operator_iff_is_compact_closure_image_closed_ball | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"closure",
"has_continuous_const_smul",
"is_compact",
"is_compact_operator",
"is_compact_operator_iff_exists_mem_nhds_is_compact_closure_image",
"metric.closed_ball",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.smul {S : Type*} [monoid S] [distrib_mul_action S M₂]
[has_continuous_const_smul S M₂] {f : M₁ → M₂}
(hf : is_compact_operator f) (c : S) :
is_compact_operator (c • f) | let ⟨K, hK, hKf⟩ := hf in ⟨c • K, hK.image $ continuous_id.const_smul c,
mem_of_superset hKf (λ x hx, smul_mem_smul_set hx)⟩ | lemma | is_compact_operator.smul | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"distrib_mul_action",
"has_continuous_const_smul",
"is_compact_operator",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.add [has_continuous_add M₂] {f g : M₁ → M₂}
(hf : is_compact_operator f) (hg : is_compact_operator g) :
is_compact_operator (f + g) | let ⟨A, hA, hAf⟩ := hf, ⟨B, hB, hBg⟩ := hg in
⟨A + B, hA.add hB, mem_of_superset (inter_mem hAf hBg) (λ x ⟨hxA, hxB⟩, set.add_mem_add hxA hxB)⟩ | lemma | is_compact_operator.add | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"has_continuous_add",
"is_compact_operator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.neg [has_continuous_neg M₄] {f : M₁ → M₄}
(hf : is_compact_operator f) : is_compact_operator (-f) | let ⟨K, hK, hKf⟩ := hf in
⟨-K, hK.neg, mem_of_superset hKf $ λ x (hx : f x ∈ K), set.neg_mem_neg.mpr hx⟩ | lemma | is_compact_operator.neg | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"has_continuous_neg",
"is_compact_operator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.sub [topological_add_group M₄] {f g : M₁ → M₄}
(hf : is_compact_operator f) (hg : is_compact_operator g) : is_compact_operator (f - g) | by rw sub_eq_add_neg; exact hf.add hg.neg | lemma | is_compact_operator.sub | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"is_compact_operator",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_operator [module R₁ M₁] [module R₄ M₄] [has_continuous_const_smul R₄ M₄]
[topological_add_group M₄] :
submodule R₄ (M₁ →SL[σ₁₄] M₄) | { carrier := {f | is_compact_operator f},
add_mem' := λ f g hf hg, hf.add hg,
zero_mem' := is_compact_operator_zero,
smul_mem' := λ c f hf, hf.smul c } | def | compact_operator | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"has_continuous_const_smul",
"is_compact_operator",
"is_compact_operator_zero",
"module",
"submodule",
"topological_add_group"
] | The submodule of compact continuous linear maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_operator.comp_clm [add_comm_monoid M₂] [module R₂ M₂] {f : M₂ → M₃}
(hf : is_compact_operator f) (g : M₁ →SL[σ₁₂] M₂) :
is_compact_operator (f ∘ g) | begin
have := g.continuous.tendsto 0,
rw map_zero at this,
rcases hf with ⟨K, hK, hKf⟩,
exact ⟨K, hK, this hKf⟩
end | lemma | is_compact_operator.comp_clm | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"add_comm_monoid",
"is_compact_operator",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.continuous_comp {f : M₁ → M₂} (hf : is_compact_operator f) {g : M₂ → M₃}
(hg : continuous g) :
is_compact_operator (g ∘ f) | begin
rcases hf with ⟨K, hK, hKf⟩,
refine ⟨g '' K, hK.image hg, mem_of_superset hKf _⟩,
nth_rewrite 1 preimage_comp,
exact preimage_mono (subset_preimage_image _ _)
end | lemma | is_compact_operator.continuous_comp | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"continuous",
"is_compact_operator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.clm_comp [add_comm_monoid M₂] [module R₂ M₂] [add_comm_monoid M₃]
[module R₃ M₃] {f : M₁ → M₂} (hf : is_compact_operator f) (g : M₂ →SL[σ₂₃] M₃) :
is_compact_operator (g ∘ f) | hf.continuous_comp g.continuous | lemma | is_compact_operator.clm_comp | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"add_comm_monoid",
"is_compact_operator",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.cod_restrict {f : M₁ → M₂} (hf : is_compact_operator f)
{V : submodule R₂ M₂} (hV : ∀ x, f x ∈ V) (h_closed : is_closed (V : set M₂)):
is_compact_operator (set.cod_restrict f V hV) | let ⟨K, hK, hKf⟩ := hf in
⟨coe ⁻¹' K, (closed_embedding_subtype_coe h_closed).is_compact_preimage hK, hKf⟩ | lemma | is_compact_operator.cod_restrict | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"closed_embedding_subtype_coe",
"is_closed",
"is_compact_operator",
"set.cod_restrict",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator.restrict {f : M₁ →ₗ[R₁] M₁} (hf : is_compact_operator f)
{V : submodule R₁ M₁} (hV : ∀ v ∈ V, f v ∈ V) (h_closed : is_closed (V : set M₁)):
is_compact_operator (f.restrict hV) | (hf.comp_clm V.subtypeL).cod_restrict (set_like.forall.2 hV) h_closed | lemma | is_compact_operator.restrict | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"is_closed",
"is_compact_operator",
"submodule"
] | If a compact operator preserves a closed submodule, its restriction to that submodule is
compact.
Note that, following mathlib's convention in linear algebra, `restrict` designates the restriction
of an endomorphism `f : E →ₗ E` to an endomorphism `f' : ↥V →ₗ ↥V`. To prove that the restriction
`f' : ↥U →ₛₗ ↥V` of a co... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_operator.restrict' [separated_space M₂] {f : M₂ →ₗ[R₂] M₂}
(hf : is_compact_operator f) {V : submodule R₂ M₂} (hV : ∀ v ∈ V, f v ∈ V)
[hcomplete : complete_space V] : is_compact_operator (f.restrict hV) | hf.restrict hV (complete_space_coe_iff_is_complete.mp hcomplete).is_closed | lemma | is_compact_operator.restrict' | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"complete_space",
"is_closed",
"is_compact_operator",
"separated_space",
"submodule"
] | If a compact operator preserves a complete submodule, its restriction to that submodule is
compact.
Note that, following mathlib's convention in linear algebra, `restrict` designates the restriction
of an endomorphism `f : E →ₗ E` to an endomorphism `f' : ↥V →ₗ ↥V`. To prove that the restriction
`f' : ↥U →ₛₗ ↥V` of a ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_operator.continuous {f : M₁ →ₛₗ[σ₁₂] M₂}
(hf : is_compact_operator f) : continuous f | begin
letI : uniform_space M₂ := topological_add_group.to_uniform_space _,
haveI : uniform_add_group M₂ := topological_add_comm_group_is_uniform,
-- Since `f` is linear, we only need to show that it is continuous at zero.
-- Let `U` be a neighborhood of `0` in `M₂`.
refine continuous_of_continuous_at_zero f (... | lemma | is_compact_operator.continuous | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"continuous",
"inv_ne_zero",
"is_compact_operator",
"is_unit",
"map_inv₀",
"map_ne_zero",
"mem_map",
"normed_field.exists_lt_norm",
"preimage_smul_setₛₗ",
"set_smul_mem_nhds_zero_iff",
"uniform_add_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.mk_of_is_compact_operator {f : M₁ →ₛₗ[σ₁₂] M₂}
(hf : is_compact_operator f) : M₁ →SL[σ₁₂] M₂ | ⟨f, hf.continuous⟩ | def | continuous_linear_map.mk_of_is_compact_operator | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"is_compact_operator"
] | Upgrade a compact `linear_map` to a `continuous_linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.mk_of_is_compact_operator_to_linear_map {f : M₁ →ₛₗ[σ₁₂] M₂}
(hf : is_compact_operator f) :
(continuous_linear_map.mk_of_is_compact_operator hf : M₁ →ₛₗ[σ₁₂] M₂) = f | rfl | lemma | continuous_linear_map.mk_of_is_compact_operator_to_linear_map | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.mk_of_is_compact_operator",
"is_compact_operator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.coe_mk_of_is_compact_operator {f : M₁ →ₛₗ[σ₁₂] M₂}
(hf : is_compact_operator f) :
(continuous_linear_map.mk_of_is_compact_operator hf : M₁ → M₂) = f | rfl | lemma | continuous_linear_map.coe_mk_of_is_compact_operator | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.mk_of_is_compact_operator",
"is_compact_operator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.mk_of_is_compact_operator_mem_compact_operator {f : M₁ →ₛₗ[σ₁₂] M₂}
(hf : is_compact_operator f) :
continuous_linear_map.mk_of_is_compact_operator hf ∈ compact_operator σ₁₂ M₁ M₂ | hf | lemma | continuous_linear_map.mk_of_is_compact_operator_mem_compact_operator | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"compact_operator",
"continuous_linear_map.mk_of_is_compact_operator",
"is_compact_operator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_set_of_is_compact_operator {𝕜₁ 𝕜₂ : Type*} [nontrivially_normed_field 𝕜₁]
[normed_field 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ M₂ : Type*} [seminormed_add_comm_group M₁]
[add_comm_group M₂] [normed_space 𝕜₁ M₁] [module 𝕜₂ M₂] [uniform_space M₂] [uniform_add_group M₂]
[has_continuous_const_smul 𝕜₂ M₂] [t2_sp... | begin
refine is_closed_of_closure_subset _,
rintros u hu,
rw [mem_closure_iff_nhds_zero] at hu,
suffices : totally_bounded (u '' metric.closed_ball 0 1),
{ change is_compact_operator (u : M₁ →ₛₗ[σ₁₂] M₂),
rw is_compact_operator_iff_is_compact_closure_image_closed_ball (u : M₁ →ₛₗ[σ₁₂] M₂)
zero_lt_on... | lemma | is_closed_set_of_is_compact_operator | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"add_comm_group",
"complete_space",
"continuous_linear_map.sub_apply",
"has_continuous_const_smul",
"is_closed",
"is_closed_closure",
"is_closed_of_closure_subset",
"is_compact_of_totally_bounded_is_closed",
"is_compact_operator",
"is_compact_operator_iff_is_compact_closure_image_closed_ball",
"... | The set of compact operators from a normed space to a complete topological vector space is
closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_operator_topological_closure {𝕜₁ 𝕜₂ : Type*} [nontrivially_normed_field 𝕜₁]
[normed_field 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ M₂ : Type*}
[seminormed_add_comm_group M₁] [add_comm_group M₂] [normed_space 𝕜₁ M₁] [module 𝕜₂ M₂]
[uniform_space M₂] [uniform_add_group M₂] [has_continuous_const_smul 𝕜₂ M₂] [t2_sp... | set_like.ext' (is_closed_set_of_is_compact_operator.closure_eq) | lemma | compact_operator_topological_closure | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"add_comm_group",
"compact_operator",
"complete_space",
"has_continuous_const_smul",
"has_continuous_smul",
"module",
"nontrivially_normed_field",
"normed_field",
"normed_space",
"seminormed_add_comm_group",
"set_like.ext'",
"t2_space",
"uniform_add_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_operator_of_tendsto {ι 𝕜₁ 𝕜₂ : Type*} [nontrivially_normed_field 𝕜₁]
[normed_field 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ M₂ : Type*}
[seminormed_add_comm_group M₁] [add_comm_group M₂] [normed_space 𝕜₁ M₁] [module 𝕜₂ M₂]
[uniform_space M₂] [uniform_add_group M₂] [has_continuous_const_smul 𝕜₂ M₂] [t2_space ... | is_closed_set_of_is_compact_operator.mem_of_tendsto hf hF | lemma | is_compact_operator_of_tendsto | analysis.normed_space | src/analysis/normed_space/compact_operator.lean | [
"analysis.locally_convex.bounded",
"topology.algebra.module.strong_topology"
] | [
"add_comm_group",
"complete_space",
"filter",
"has_continuous_const_smul",
"is_compact_operator",
"module",
"nontrivially_normed_field",
"normed_field",
"normed_space",
"seminormed_add_comm_group",
"t2_space",
"uniform_add_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_closed_complemented_of_finite_dimensional_range (f : E →L[𝕜] F)
[finite_dimensional 𝕜 (range f)] :
(ker f).closed_complemented | begin
set f' : E →L[𝕜] (range f) := f.cod_restrict _ (f : E →ₗ[𝕜] F).mem_range_self,
rcases f'.exists_right_inverse_of_surjective (f : E →ₗ[𝕜] F).range_range_restrict with ⟨g, hg⟩,
simpa only [ker_cod_restrict] using f'.closed_complemented_ker_of_right_inverse g (ext_iff.1 hg)
end | lemma | continuous_linear_map.ker_closed_complemented_of_finite_dimensional_range | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"finite_dimensional"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_prod_of_surjective_of_is_compl (f : E →L[𝕜] F) (g : E →L[𝕜] G) (hf : range f = ⊤)
(hg : range g = ⊤) (hfg : is_compl (ker f) (ker g)) :
E ≃L[𝕜] F × G | ((f : E →ₗ[𝕜] F).equiv_prod_of_surjective_of_is_compl ↑g hf hg
hfg).to_continuous_linear_equiv_of_continuous (f.continuous.prod_mk g.continuous) | def | continuous_linear_map.equiv_prod_of_surjective_of_is_compl | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"is_compl"
] | If `f : E →L[R] F` and `g : E →L[R] G` are two surjective linear maps and
their kernels are complement of each other, then `x ↦ (f x, g x)` defines
a linear equivalence `E ≃L[R] F × G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_equiv_prod_of_surjective_of_is_compl {f : E →L[𝕜] F} {g : E →L[𝕜] G}
(hf : range f = ⊤) (hg : range g = ⊤) (hfg : is_compl (ker f) (ker g)) :
(equiv_prod_of_surjective_of_is_compl f g hf hg hfg : E →ₗ[𝕜] F × G) = f.prod g | rfl | lemma | continuous_linear_map.coe_equiv_prod_of_surjective_of_is_compl | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_prod_of_surjective_of_is_compl_to_linear_equiv {f : E →L[𝕜] F} {g : E →L[𝕜] G}
(hf : range f = ⊤) (hg : range g = ⊤) (hfg : is_compl (ker f) (ker g)) :
(equiv_prod_of_surjective_of_is_compl f g hf hg hfg).to_linear_equiv =
linear_map.equiv_prod_of_surjective_of_is_compl f g hf hg hfg | rfl | lemma | continuous_linear_map.equiv_prod_of_surjective_of_is_compl_to_linear_equiv | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"is_compl",
"linear_map.equiv_prod_of_surjective_of_is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_prod_of_surjective_of_is_compl_apply {f : E →L[𝕜] F} {g : E →L[𝕜] G}
(hf : range f = ⊤) (hg : range g = ⊤) (hfg : is_compl (ker f) (ker g)) (x : E) :
equiv_prod_of_surjective_of_is_compl f g hf hg hfg x = (f x, g x) | rfl | lemma | continuous_linear_map.equiv_prod_of_surjective_of_is_compl_apply | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_equiv_of_closed_compl (h : is_compl p q) (hp : is_closed (p : set E))
(hq : is_closed (q : set E)) : (p × q) ≃L[𝕜] E | begin
haveI := hp.complete_space_coe, haveI := hq.complete_space_coe,
refine (p.prod_equiv_of_is_compl q h).to_continuous_linear_equiv_of_continuous _,
exact (p.subtypeL.coprod q.subtypeL).continuous
end | def | subspace.prod_equiv_of_closed_compl | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"continuous",
"is_closed",
"is_compl"
] | If `q` is a closed complement of a closed subspace `p`, then `p × q` is continuously
isomorphic to `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_proj_of_closed_compl (h : is_compl p q) (hp : is_closed (p : set E))
(hq : is_closed (q : set E)) :
E →L[𝕜] p | (continuous_linear_map.fst 𝕜 p q) ∘L ↑(prod_equiv_of_closed_compl p q h hp hq).symm | def | subspace.linear_proj_of_closed_compl | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"continuous_linear_map.fst",
"is_closed",
"is_compl"
] | Projection to a closed submodule along a closed complement. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod_equiv_of_closed_compl (h : is_compl p q) (hp : is_closed (p : set E))
(hq : is_closed (q : set E)) :
⇑(p.prod_equiv_of_closed_compl q h hp hq) = p.prod_equiv_of_is_compl q h | rfl | lemma | subspace.coe_prod_equiv_of_closed_compl | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"is_closed",
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_prod_equiv_of_closed_compl_symm (h : is_compl p q) (hp : is_closed (p : set E))
(hq : is_closed (q : set E)) :
⇑(p.prod_equiv_of_closed_compl q h hp hq).symm = (p.prod_equiv_of_is_compl q h).symm | rfl | lemma | subspace.coe_prod_equiv_of_closed_compl_symm | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"is_closed",
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_continuous_linear_proj_of_closed_compl (h : is_compl p q)
(hp : is_closed (p : set E)) (hq : is_closed (q : set E)) :
(p.linear_proj_of_closed_compl q h hp hq : E →ₗ[𝕜] p) = p.linear_proj_of_is_compl q h | rfl | lemma | subspace.coe_continuous_linear_proj_of_closed_compl | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"is_closed",
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_continuous_linear_proj_of_closed_compl' (h : is_compl p q)
(hp : is_closed (p : set E)) (hq : is_closed (q : set E)) :
⇑(p.linear_proj_of_closed_compl q h hp hq) = p.linear_proj_of_is_compl q h | rfl | lemma | subspace.coe_continuous_linear_proj_of_closed_compl' | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"is_closed",
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_complemented_of_closed_compl (h : is_compl p q) (hp : is_closed (p : set E))
(hq : is_closed (q : set E)) : p.closed_complemented | ⟨p.linear_proj_of_closed_compl q h hp hq, submodule.linear_proj_of_is_compl_apply_left h⟩ | lemma | subspace.closed_complemented_of_closed_compl | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"is_closed",
"is_compl",
"submodule.linear_proj_of_is_compl_apply_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_complemented_iff_has_closed_compl : p.closed_complemented ↔
is_closed (p : set E) ∧ ∃ (q : subspace 𝕜 E) (hq : is_closed (q : set E)), is_compl p q | ⟨λ h, ⟨h.is_closed, h.has_closed_complement⟩,
λ ⟨hp, ⟨q, hq, hpq⟩⟩, closed_complemented_of_closed_compl hpq hp hq⟩ | lemma | subspace.closed_complemented_iff_has_closed_compl | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"is_closed",
"is_compl",
"subspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_complemented_of_quotient_finite_dimensional [complete_space 𝕜]
[finite_dimensional 𝕜 (E ⧸ p)] (hp : is_closed (p : set E)) :
p.closed_complemented | begin
obtain ⟨q, hq⟩ : ∃ q, is_compl p q := p.exists_is_compl,
haveI : finite_dimensional 𝕜 q := (p.quotient_equiv_of_is_compl q hq).finite_dimensional,
exact closed_complemented_of_closed_compl hq hp q.closed_of_finite_dimensional
end | lemma | subspace.closed_complemented_of_quotient_finite_dimensional | analysis.normed_space | src/analysis/normed_space/complemented.lean | [
"analysis.normed_space.banach",
"analysis.normed_space.finite_dimension"
] | [
"complete_space",
"finite_dimensional",
"is_closed",
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_space.to_has_uniform_continuous_const_smul :
has_uniform_continuous_const_smul 𝕜 E | ⟨λ c, (lipschitz_with_smul c).uniform_continuous⟩ | instance | uniform_space.completion.normed_space.to_has_uniform_continuous_const_smul | analysis.normed_space | src/analysis/normed_space/completion.lean | [
"analysis.normed.group.completion",
"analysis.normed_space.operator_norm",
"topology.algebra.uniform_ring"
] | [
"has_uniform_continuous_const_smul",
"lipschitz_with_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_complₗᵢ : E →ₗᵢ[𝕜] completion E | { to_fun := coe,
map_smul' := coe_smul,
norm_map' := norm_coe,
.. to_compl } | def | uniform_space.completion.to_complₗᵢ | analysis.normed_space | src/analysis/normed_space/completion.lean | [
"analysis.normed.group.completion",
"analysis.normed_space.operator_norm",
"topology.algebra.uniform_ring"
] | [] | Embedding of a normed space to its completion as a linear isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_complₗᵢ : ⇑(to_complₗᵢ : E →ₗᵢ[𝕜] completion E) = coe | rfl | lemma | uniform_space.completion.coe_to_complₗᵢ | analysis.normed_space | src/analysis/normed_space/completion.lean | [
"analysis.normed.group.completion",
"analysis.normed_space.operator_norm",
"topology.algebra.uniform_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_complL : E →L[𝕜] completion E | to_complₗᵢ.to_continuous_linear_map | def | uniform_space.completion.to_complL | analysis.normed_space | src/analysis/normed_space/completion.lean | [
"analysis.normed.group.completion",
"analysis.normed_space.operator_norm",
"topology.algebra.uniform_ring"
] | [] | Embedding of a normed space to its completion as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_complL : ⇑(to_complL : E →L[𝕜] completion E) = coe | rfl | lemma | uniform_space.completion.coe_to_complL | analysis.normed_space | src/analysis/normed_space/completion.lean | [
"analysis.normed.group.completion",
"analysis.normed_space.operator_norm",
"topology.algebra.uniform_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_to_complL {𝕜 E : Type*} [nontrivially_normed_field 𝕜] [normed_add_comm_group E]
[normed_space 𝕜 E] [nontrivial E] : ‖(to_complL : E →L[𝕜] completion E)‖ = 1 | (to_complₗᵢ : E →ₗᵢ[𝕜] completion E).norm_to_continuous_linear_map | lemma | uniform_space.completion.norm_to_complL | analysis.normed_space | src/analysis/normed_space/completion.lean | [
"analysis.normed.group.completion",
"analysis.normed_space.operator_norm",
"topology.algebra.uniform_ring"
] | [
"nontrivial",
"nontrivially_normed_field",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conformal_map {R : Type*} {X Y : Type*} [normed_field R]
[seminormed_add_comm_group X] [seminormed_add_comm_group Y] [normed_space R X] [normed_space R Y]
(f' : X →L[R] Y) | ∃ (c : R) (hc : c ≠ 0) (li : X →ₗᵢ[R] Y), f' = c • li.to_continuous_linear_map | def | is_conformal_map | analysis.normed_space | src/analysis/normed_space/conformal_linear_map.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry"
] | [
"normed_field",
"normed_space",
"seminormed_add_comm_group"
] | A continuous linear map `f'` is said to be conformal if it's
a nonzero multiple of a linear isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_conformal_map_id : is_conformal_map (id R M) | ⟨1, one_ne_zero, id, by simp⟩ | lemma | is_conformal_map_id | analysis.normed_space | src/analysis/normed_space/conformal_linear_map.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry"
] | [
"is_conformal_map",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conformal_map.smul (hf : is_conformal_map f) {c : R} (hc : c ≠ 0) :
is_conformal_map (c • f) | begin
rcases hf with ⟨c', hc', li, rfl⟩,
exact ⟨c * c', mul_ne_zero hc hc', li, smul_smul _ _ _⟩
end | lemma | is_conformal_map.smul | 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",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conformal_map_const_smul (hc : c ≠ 0) : is_conformal_map (c • id R M) | is_conformal_map_id.smul hc | lemma | is_conformal_map_const_smul | analysis.normed_space | src/analysis/normed_space/conformal_linear_map.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry"
] | [
"is_conformal_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_isometry.is_conformal_map (f' : M →ₗᵢ[R] N) :
is_conformal_map f'.to_continuous_linear_map | ⟨1, one_ne_zero, f', (one_smul _ _).symm⟩ | lemma | linear_isometry.is_conformal_map | analysis.normed_space | src/analysis/normed_space/conformal_linear_map.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry"
] | [
"is_conformal_map",
"one_ne_zero",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conformal_map_of_subsingleton [subsingleton M] (f' : M →L[R] N) :
is_conformal_map f' | ⟨1, one_ne_zero, ⟨0, λ x, by simp [subsingleton.elim x 0]⟩, subsingleton.elim _ _⟩ | lemma | is_conformal_map_of_subsingleton | analysis.normed_space | src/analysis/normed_space/conformal_linear_map.lean | [
"analysis.normed_space.basic",
"analysis.normed_space.linear_isometry"
] | [
"is_conformal_map",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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