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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map_midpoint (f : PE ≃ᵢ PF) (x y : PE) : f (midpoint ℝ x y) = midpoint ℝ (f x) (f y) | begin
set e : PE ≃ᵢ PE :=
((f.trans $ (point_reflection ℝ $ midpoint ℝ (f x) (f y)).to_isometry_equiv).trans f.symm).trans
(point_reflection ℝ $ midpoint ℝ x y).to_isometry_equiv,
have hx : e x = x, by simp,
have hy : e y = y, by simp,
have hm := e.midpoint_fixed hx hy,
simp only [e, trans_apply] at h... | lemma | isometry_equiv.map_midpoint | analysis.normed_space | src/analysis/normed_space/mazur_ulam.lean | [
"topology.instances.real_vector_space",
"analysis.normed_space.affine_isometry"
] | [
"midpoint"
] | A bijective isometry sends midpoints to midpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_real_linear_isometry_equiv_of_map_zero (f : E ≃ᵢ F) (h0 : f 0 = 0) :
E ≃ₗᵢ[ℝ] F | { norm_map' := λ x, show ‖f x‖ = ‖x‖, by simp only [← dist_zero_right, ← h0, f.dist_eq],
.. ((add_monoid_hom.of_map_midpoint ℝ ℝ f h0 f.map_midpoint).to_real_linear_map f.continuous),
.. f } | def | isometry_equiv.to_real_linear_isometry_equiv_of_map_zero | analysis.normed_space | src/analysis/normed_space/mazur_ulam.lean | [
"topology.instances.real_vector_space",
"analysis.normed_space.affine_isometry"
] | [
"add_monoid_hom.of_map_midpoint"
] | **Mazur-Ulam Theorem**: if `f` is an isometric bijection between two normed vector spaces
over `ℝ` and `f 0 = 0`, then `f` is a linear isometry equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_real_linear_equiv_of_map_zero (f : E ≃ᵢ F) (h0 : f 0 = 0) :
⇑(f.to_real_linear_isometry_equiv_of_map_zero h0) = f | rfl | lemma | isometry_equiv.coe_to_real_linear_equiv_of_map_zero | analysis.normed_space | src/analysis/normed_space/mazur_ulam.lean | [
"topology.instances.real_vector_space",
"analysis.normed_space.affine_isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_real_linear_equiv_of_map_zero_symm (f : E ≃ᵢ F) (h0 : f 0 = 0) :
⇑(f.to_real_linear_isometry_equiv_of_map_zero h0).symm = f.symm | rfl | lemma | isometry_equiv.coe_to_real_linear_equiv_of_map_zero_symm | analysis.normed_space | src/analysis/normed_space/mazur_ulam.lean | [
"topology.instances.real_vector_space",
"analysis.normed_space.affine_isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_linear_isometry_equiv (f : E ≃ᵢ F) : E ≃ₗᵢ[ℝ] F | (f.trans (isometry_equiv.add_right (f 0)).symm).to_real_linear_isometry_equiv_of_map_zero
(by simpa only [sub_eq_add_neg] using sub_self (f 0)) | def | isometry_equiv.to_real_linear_isometry_equiv | analysis.normed_space | src/analysis/normed_space/mazur_ulam.lean | [
"topology.instances.real_vector_space",
"analysis.normed_space.affine_isometry"
] | [] | **Mazur-Ulam Theorem**: if `f` is an isometric bijection between two normed vector spaces
over `ℝ`, then `x ↦ f x - f 0` is a linear isometry equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_real_linear_equiv_apply (f : E ≃ᵢ F) (x : E) :
(f.to_real_linear_isometry_equiv : E → F) x = f x - f 0 | (sub_eq_add_neg (f x) (f 0)).symm | lemma | isometry_equiv.to_real_linear_equiv_apply | analysis.normed_space | src/analysis/normed_space/mazur_ulam.lean | [
"topology.instances.real_vector_space",
"analysis.normed_space.affine_isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_linear_isometry_equiv_symm_apply (f : E ≃ᵢ F) (y : F) :
(f.to_real_linear_isometry_equiv.symm : F → E) y = f.symm (y + f 0) | rfl | lemma | isometry_equiv.to_real_linear_isometry_equiv_symm_apply | analysis.normed_space | src/analysis/normed_space/mazur_ulam.lean | [
"topology.instances.real_vector_space",
"analysis.normed_space.affine_isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_affine_isometry_equiv (f : PE ≃ᵢ PF) : PE ≃ᵃⁱ[ℝ] PF | affine_isometry_equiv.mk' f
(((vadd_const (classical.arbitrary PE)).trans $ f.trans
(vadd_const (f $ classical.arbitrary PE)).symm).to_real_linear_isometry_equiv)
(classical.arbitrary PE) (λ p, by simp) | def | isometry_equiv.to_real_affine_isometry_equiv | analysis.normed_space | src/analysis/normed_space/mazur_ulam.lean | [
"topology.instances.real_vector_space",
"analysis.normed_space.affine_isometry"
] | [
"affine_isometry_equiv.mk'",
"classical.arbitrary"
] | **Mazur-Ulam Theorem**: if `f` is an isometric bijection between two normed add-torsors over
normed vector spaces over `ℝ`, then `f` is an affine isometry equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_to_real_affine_isometry_equiv (f : PE ≃ᵢ PF) :
⇑f.to_real_affine_isometry_equiv = f | rfl | lemma | isometry_equiv.coe_fn_to_real_affine_isometry_equiv | analysis.normed_space | src/analysis/normed_space/mazur_ulam.lean | [
"topology.instances.real_vector_space",
"analysis.normed_space.affine_isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_real_affine_isometry_equiv (f : PE ≃ᵢ PF) :
f.to_real_affine_isometry_equiv.to_isometry_equiv = f | by { ext, refl } | lemma | isometry_equiv.coe_to_real_affine_isometry_equiv | analysis.normed_space | src/analysis/normed_space/mazur_ulam.lean | [
"topology.instances.real_vector_space",
"analysis.normed_space.affine_isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bound_of_shell {ε : ι → ℝ} {C : ℝ} (hε : ∀ i, 0 < ε i) {c : ι → 𝕜} (hc : ∀ i, 1 < ‖c i‖)
(hf : ∀ m : Π i, E i, (∀ i, ε i / ‖c i‖ ≤ ‖m i‖) → (∀ i, ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖)
(m : Π i, E i) : ‖f m‖ ≤ C * ∏ i, ‖m i‖ | begin
rcases em (∃ i, m i = 0) with ⟨i, hi⟩|hm; [skip, push_neg at hm],
{ simp [f.map_coord_zero i hi, prod_eq_zero (mem_univ i), hi] },
choose δ hδ0 hδm_lt hle_δm hδinv using λ i, rescale_to_shell (hc i) (hε i) (hm i),
have hδ0 : 0 < ∏ i, ‖δ i‖, from prod_pos (λ i _, norm_pos_iff.2 (hδ0 i)),
simpa [map_smul_... | lemma | multilinear_map.bound_of_shell | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"em",
"mul_le_mul_left",
"mul_left_comm",
"norm_smul",
"rescale_to_shell"
] | If a multilinear map in finitely many variables on normed spaces satisfies the inequality
`‖f m‖ ≤ C * ∏ i, ‖m i‖` on a shell `ε i / ‖c i‖ < ‖m i‖ < ε i` for some positive numbers `ε i`
and elements `c i : 𝕜`, `1 < ‖c i‖`, then it satisfies this inequality for all `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_bound_of_continuous (hf : continuous f) :
∃ (C : ℝ), 0 < C ∧ (∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) | begin
casesI is_empty_or_nonempty ι,
{ refine ⟨‖f 0‖ + 1, add_pos_of_nonneg_of_pos (norm_nonneg _) zero_lt_one, λ m, _⟩,
obtain rfl : m = 0, from funext (is_empty.elim ‹_›),
simp [univ_eq_empty, zero_le_one] },
obtain ⟨ε : ℝ, ε0 : 0 < ε, hε : ∀ m : Π i, E i, ‖m - 0‖ < ε → ‖f m - f 0‖ < 1⟩ :=
normed_ad... | theorem | multilinear_map.exists_bound_of_continuous | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous",
"div_le_iff'",
"div_nonneg",
"div_pos",
"fintype.card",
"inv_div",
"inv_pow",
"is_empty.elim",
"is_empty_or_nonempty",
"normed_field.exists_one_lt_norm",
"one_div",
"pow_pos",
"zero_le_one",
"zero_lt_one"
] | If a multilinear map in finitely many variables on normed spaces is continuous, then it
satisfies the inequality `‖f m‖ ≤ C * ∏ i, ‖m i‖`, for some `C` which can be chosen to be
positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_bound' [decidable_eq ι] {C : ℝ} (hC : 0 ≤ C)
(H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) (m₁ m₂ : Πi, E i) :
‖f m₁ - f m₂‖ ≤
C * ∑ i, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ | begin
have A : ∀(s : finset ι), ‖f m₁ - f (s.piecewise m₂ m₁)‖
≤ C * ∑ i in s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖,
{ refine finset.induction (by simp) _,
assume i s his Hrec,
have I : ‖f (s.piecewise m₂ m₁) - f ((insert i s).piecewise m₂ m₁)‖
≤ C * ∏ j, if j = i then ‖m₁ i - m... | lemma | multilinear_map.norm_image_sub_le_of_bound' | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"dist_triangle",
"finset",
"finset.induction",
"left_distrib",
"mul_le_mul_of_nonneg_left"
] | If `f` satisfies a boundedness property around `0`, one can deduce a bound on `f m₁ - f m₂`
using the multilinearity. Here, we give a precise but hard to use version. See
`norm_image_sub_le_of_bound` for a less precise but more usable version. The bound reads
`‖f m - f m'‖ ≤
C * ‖m 1 - m' 1‖ * max ‖m 2‖ ‖m' 2‖ * max ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le_of_bound {C : ℝ} (hC : 0 ≤ C)
(H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) (m₁ m₂ : Πi, E i) :
‖f m₁ - f m₂‖ ≤ C * (fintype.card ι) * (max ‖m₁‖ ‖m₂‖) ^ (fintype.card ι - 1) * ‖m₁ - m₂‖ | begin
letI := classical.dec_eq ι,
have A : ∀ (i : ι), ∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖)
≤ ‖m₁ - m₂‖ * (max ‖m₁‖ ‖m₂‖) ^ (fintype.card ι - 1),
{ assume i,
calc ∏ j, (if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖)
≤ ∏ j : ι, function.update (λ j, max ‖m₁‖ ‖m₂‖) i (‖m₁ - m₂‖)... | lemma | multilinear_map.norm_image_sub_le_of_bound | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"classical.dec_eq",
"finset.mem_univ",
"fintype.card",
"max_le_max",
"mul_le_mul_of_nonneg_left",
"nsmul_eq_mul",
"ring"
] | If `f` satisfies a boundedness property around `0`, one can deduce a bound on `f m₁ - f m₂`
using the multilinearity. Here, we give a usable but not very precise version. See
`norm_image_sub_le_of_bound'` for a more precise but less usable version. The bound is
`‖f m - f m'‖ ≤ C * card ι * ‖m - m'‖ * (max ‖m‖ ‖m'‖) ^ (... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_of_bound (C : ℝ) (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) :
continuous f | begin
let D := max C 1,
have D_pos : 0 ≤ D := le_trans zero_le_one (le_max_right _ _),
replace H : ∀ m, ‖f m‖ ≤ D * ∏ i, ‖m i‖,
{ assume m,
apply le_trans (H m) (mul_le_mul_of_nonneg_right (le_max_left _ _) _),
exact prod_nonneg (λ(i : ι) hi, norm_nonneg (m i)) },
refine continuous_iff_continuous_at.2... | theorem | multilinear_map.continuous_of_bound | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous",
"continuous_at_of_locally_lipschitz",
"fintype.card",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"nat.cast_nonneg",
"pow_le_pow_of_le_left",
"zero_le_one",
"zero_lt_one"
] | If a multilinear map satisfies an inequality `‖f m‖ ≤ C * ∏ i, ‖m i‖`, then it is
continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_continuous (C : ℝ) (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) :
continuous_multilinear_map 𝕜 E G | { cont := f.continuous_of_bound C H, ..f } | def | multilinear_map.mk_continuous | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"cont",
"continuous_multilinear_map"
] | Constructing a continuous multilinear map from a multilinear map satisfying a boundedness
condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk_continuous (C : ℝ) (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) :
⇑(f.mk_continuous C H) = f | rfl | lemma | multilinear_map.coe_mk_continuous | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restr_norm_le {k n : ℕ} (f : (multilinear_map 𝕜 (λ i : fin n, G) G' : _))
(s : finset (fin n)) (hk : s.card = k) (z : G) {C : ℝ}
(H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) (v : fin k → G) :
‖f.restr s hk z v‖ ≤ C * ‖z‖ ^ (n - k) * ∏ i, ‖v i‖ | begin
rw [mul_right_comm, mul_assoc],
convert H _ using 2,
simp only [apply_dite norm, fintype.prod_dite, prod_const (‖z‖), finset.card_univ,
fintype.card_of_subtype sᶜ (λ x, mem_compl), card_compl, fintype.card_fin, hk, mk_coe,
← (s.order_iso_of_fin hk).symm.bijective.prod_comp (λ x, ‖v x‖)],
refl
end | lemma | multilinear_map.restr_norm_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"apply_dite",
"finset",
"finset.card_univ",
"fintype.card_fin",
"fintype.card_of_subtype",
"fintype.prod_dite",
"mul_assoc",
"mul_right_comm",
"multilinear_map"
] | Given a multilinear map in `n` variables, if one restricts it to `k` variables putting `z` on
the other coordinates, then the resulting restricted function satisfies an inequality
`‖f.restr v‖ ≤ C * ‖z‖^(n-k) * Π ‖v i‖` if the original function satisfies `‖f v‖ ≤ C * Π ‖v i‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bound : ∃ (C : ℝ), 0 < C ∧ (∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) | f.to_multilinear_map.exists_bound_of_continuous f.2 | theorem | continuous_multilinear_map.bound | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"bound"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm | Inf {c | 0 ≤ (c : ℝ) ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖} | def | continuous_multilinear_map.op_norm | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [] | The operator norm of a continuous multilinear map is the inf of all its bounds. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_op_norm : has_norm (continuous_multilinear_map 𝕜 E G) | ⟨op_norm⟩ | instance | continuous_multilinear_map.has_op_norm | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"has_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_op_norm' : has_norm (continuous_multilinear_map 𝕜 (λ (i : ι), G) G') | continuous_multilinear_map.has_op_norm | instance | continuous_multilinear_map.has_op_norm' | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"continuous_multilinear_map.has_op_norm",
"has_norm"
] | An alias of `continuous_multilinear_map.has_op_norm` with non-dependent types to help typeclass
search. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_def : ‖f‖ = Inf {c | 0 ≤ (c : ℝ) ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖} | rfl | lemma | continuous_multilinear_map.norm_def | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounds_nonempty {f : continuous_multilinear_map 𝕜 E G} :
∃ c, c ∈ {c | 0 ≤ c ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖} | let ⟨M, hMp, hMb⟩ := f.bound in ⟨M, le_of_lt hMp, hMb⟩ | lemma | continuous_multilinear_map.bounds_nonempty | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounds_bdd_below {f : continuous_multilinear_map 𝕜 E G} :
bdd_below {c | 0 ≤ c ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖} | ⟨0, λ _ ⟨hn, _⟩, hn⟩ | lemma | continuous_multilinear_map.bounds_bdd_below | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"bdd_below",
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_op_norm : ‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖ | begin
have A : 0 ≤ ∏ i, ‖m i‖ := prod_nonneg (λj hj, norm_nonneg _),
cases A.eq_or_lt with h hlt,
{ rcases prod_eq_zero_iff.1 h.symm with ⟨i, _, hi⟩,
rw norm_eq_zero at hi,
have : f m = 0 := f.map_coord_zero i hi,
rw [this, norm_zero],
exact mul_nonneg (op_norm_nonneg f) A },
{ rw [← div_le_iff ... | theorem | continuous_multilinear_map.le_op_norm | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"div_le_iff",
"le_cInf",
"norm_eq_zero"
] | The fundamental property of the operator norm of a continuous multilinear map:
`‖f m‖` is bounded by `‖f‖` times the product of the `‖m i‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_of_op_norm_le {C : ℝ} (h : ‖f‖ ≤ C) : ‖f m‖ ≤ C * ∏ i, ‖m i‖ | (f.le_op_norm m).trans $ mul_le_mul_of_nonneg_right h (prod_nonneg $ λ i _, norm_nonneg (m i)) | theorem | continuous_multilinear_map.le_of_op_norm_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ratio_le_op_norm : ‖f m‖ / ∏ i, ‖m i‖ ≤ ‖f‖ | div_le_of_nonneg_of_le_mul (prod_nonneg $ λ i _, norm_nonneg _) (op_norm_nonneg _) (f.le_op_norm m) | lemma | continuous_multilinear_map.ratio_le_op_norm | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"div_le_of_nonneg_of_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_le_op_norm (h : ‖m‖ ≤ 1) : ‖f m‖ ≤ ‖f‖ | calc
‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖ : f.le_op_norm m
... ≤ ‖f‖ * ∏ i : ι, 1 :
mul_le_mul_of_nonneg_left (prod_le_prod (λi hi, norm_nonneg _)
(λi hi, le_trans (norm_le_pi_norm (_ : Π i, E i) _) h)) (op_norm_nonneg f)
... = ‖f‖ : by simp | lemma | continuous_multilinear_map.unit_le_op_norm | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"mul_le_mul_of_nonneg_left"
] | The image of the unit ball under a continuous multilinear map is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_le_bound {M : ℝ} (hMp: 0 ≤ M) (hM : ∀ m, ‖f m‖ ≤ M * ∏ i, ‖m i‖) :
‖f‖ ≤ M | cInf_le bounds_bdd_below ⟨hMp, hM⟩ | lemma | continuous_multilinear_map.op_norm_le_bound | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"cInf_le"
] | If one controls the norm of every `f x`, then one controls the norm of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_add_le : ‖f + g‖ ≤ ‖f‖ + ‖g‖ | cInf_le bounds_bdd_below
⟨add_nonneg (op_norm_nonneg _) (op_norm_nonneg _), λ x, by { rw add_mul,
exact norm_add_le_of_le (le_op_norm _ _) (le_op_norm _ _) }⟩ | theorem | continuous_multilinear_map.op_norm_add_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"cInf_le"
] | The operator norm satisfies the triangle inequality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_zero : ‖(0 : continuous_multilinear_map 𝕜 E G)‖ = 0 | (op_norm_nonneg _).antisymm' $ op_norm_le_bound 0 le_rfl $ λ m, by simp | lemma | continuous_multilinear_map.op_norm_zero | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"antisymm'",
"continuous_multilinear_map",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_zero_iff : ‖f‖ = 0 ↔ f = 0 | ⟨λ h, by { ext m, simpa [h] using f.le_op_norm m }, by { rintro rfl, exact op_norm_zero }⟩ | theorem | continuous_multilinear_map.op_norm_zero_iff | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [] | A continuous linear map is zero iff its norm vanishes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_smul_le (c : 𝕜') : ‖c • f‖ ≤ ‖c‖ * ‖f‖ | (c • f).op_norm_le_bound
(mul_nonneg (norm_nonneg _) (op_norm_nonneg _))
begin
intro m,
erw [norm_smul, mul_assoc],
exact mul_le_mul_of_nonneg_left (le_op_norm _ _) (norm_nonneg _)
end | lemma | continuous_multilinear_map.op_norm_smul_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"mul_assoc",
"mul_le_mul_of_nonneg_left",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_neg : ‖-f‖ = ‖f‖ | by { rw norm_def, apply congr_arg, ext, simp } | lemma | continuous_multilinear_map.op_norm_neg | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_comm_group : normed_add_comm_group (continuous_multilinear_map 𝕜 E G) | add_group_norm.to_normed_add_comm_group
{ to_fun := norm,
map_zero' := op_norm_zero,
neg' := op_norm_neg,
add_le' := op_norm_add_le,
eq_zero_of_map_eq_zero' := λ f, f.op_norm_zero_iff.1 } | instance | continuous_multilinear_map.normed_add_comm_group | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"normed_add_comm_group"
] | Continuous multilinear maps themselves form a normed space with respect to
the operator norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_add_comm_group' :
normed_add_comm_group (continuous_multilinear_map 𝕜 (λ i : ι, G) G') | continuous_multilinear_map.normed_add_comm_group | instance | continuous_multilinear_map.normed_add_comm_group' | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"continuous_multilinear_map.normed_add_comm_group",
"normed_add_comm_group"
] | An alias of `continuous_multilinear_map.normed_add_comm_group` with non-dependent types to help
typeclass search. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_space : normed_space 𝕜' (continuous_multilinear_map 𝕜 E G) | ⟨λ c f, f.op_norm_smul_le c⟩ | instance | continuous_multilinear_map.normed_space | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_space' : normed_space 𝕜' (continuous_multilinear_map 𝕜 (λ i : ι, G') G) | continuous_multilinear_map.normed_space | instance | continuous_multilinear_map.normed_space' | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"continuous_multilinear_map.normed_space",
"normed_space"
] | An alias of `continuous_multilinear_map.normed_space` with non-dependent types to help typeclass
search. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_op_norm_mul_prod_of_le {b : ι → ℝ} (hm : ∀ i, ‖m i‖ ≤ b i) : ‖f m‖ ≤ ‖f‖ * ∏ i, b i | (f.le_op_norm m).trans $ mul_le_mul_of_nonneg_left
(prod_le_prod (λ _ _, norm_nonneg _) (λ i _, hm i)) (norm_nonneg f) | theorem | continuous_multilinear_map.le_op_norm_mul_prod_of_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_op_norm_mul_pow_card_of_le {b : ℝ} (hm : ∀ i, ‖m i‖ ≤ b) :
‖f m‖ ≤ ‖f‖ * b ^ fintype.card ι | by simpa only [prod_const] using f.le_op_norm_mul_prod_of_le m hm | theorem | continuous_multilinear_map.le_op_norm_mul_pow_card_of_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"fintype.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_op_norm_mul_pow_of_le {Ei : fin n → Type*} [Π i, normed_add_comm_group (Ei i)]
[Π i, normed_space 𝕜 (Ei i)] (f : continuous_multilinear_map 𝕜 Ei G) (m : Π i, Ei i)
{b : ℝ} (hm : ‖m‖ ≤ b) :
‖f m‖ ≤ ‖f‖ * b ^ n | by simpa only [fintype.card_fin]
using f.le_op_norm_mul_pow_card_of_le m (λ i, (norm_le_pi_norm m i).trans hm) | theorem | continuous_multilinear_map.le_op_norm_mul_pow_of_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"fintype.card_fin",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_op_nnnorm : ‖f m‖₊ ≤ ‖f‖₊ * ∏ i, ‖m i‖₊ | nnreal.coe_le_coe.1 $ by { push_cast, exact f.le_op_norm m } | theorem | continuous_multilinear_map.le_op_nnnorm | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [] | The fundamental property of the operator norm of a continuous multilinear map:
`‖f m‖` is bounded by `‖f‖` times the product of the `‖m i‖`, `nnnorm` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_of_op_nnnorm_le {C : ℝ≥0} (h : ‖f‖₊ ≤ C) : ‖f m‖₊ ≤ C * ∏ i, ‖m i‖₊ | (f.le_op_nnnorm m).trans $ mul_le_mul' h le_rfl | theorem | continuous_multilinear_map.le_of_op_nnnorm_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"le_rfl",
"mul_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_prod (f : continuous_multilinear_map 𝕜 E G) (g : continuous_multilinear_map 𝕜 E G') :
‖f.prod g‖ = max (‖f‖) (‖g‖) | le_antisymm
(op_norm_le_bound _ (norm_nonneg (f, g)) (λ m,
have H : 0 ≤ ∏ i, ‖m i‖, from prod_nonneg $ λ _ _, norm_nonneg _,
by simpa only [prod_apply, prod.norm_def, max_mul_of_nonneg, H]
using max_le_max (f.le_op_norm m) (g.le_op_norm m))) $
max_le
(f.op_norm_le_bound (norm_nonneg _) $ λ m, (le... | lemma | continuous_multilinear_map.op_norm_prod | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"max_le_max",
"max_mul_of_nonneg",
"prod.norm_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_pi {ι' : Type v'} [fintype ι'] {E' : ι' → Type wE'} [Π i', normed_add_comm_group (E' i')]
[Π i', normed_space 𝕜 (E' i')] (f : Π i', continuous_multilinear_map 𝕜 E (E' i')) :
‖pi f‖ = ‖f‖ | begin
apply le_antisymm,
{ refine (op_norm_le_bound _ (norm_nonneg f) (λ m, _)),
dsimp,
rw pi_norm_le_iff_of_nonneg,
exacts [λ i, (f i).le_of_op_norm_le m (norm_le_pi_norm f i),
mul_nonneg (norm_nonneg f) (prod_nonneg $ λ _ _, norm_nonneg _)] },
{ refine (pi_norm_le_iff_of_nonneg (norm_nonneg _)... | lemma | continuous_multilinear_map.norm_pi | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"fintype",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_of_subsingleton_le [subsingleton ι] (i' : ι) : ‖of_subsingleton 𝕜 G i'‖ ≤ 1 | op_norm_le_bound _ zero_le_one $ λ m,
by rw [fintype.prod_subsingleton _ i', one_mul, of_subsingleton_apply] | lemma | continuous_multilinear_map.norm_of_subsingleton_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"fintype.prod_subsingleton",
"one_mul",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_of_subsingleton [subsingleton ι] [nontrivial G] (i' : ι) :
‖of_subsingleton 𝕜 G i'‖ = 1 | begin
apply le_antisymm (norm_of_subsingleton_le 𝕜 G i'),
obtain ⟨g, hg⟩ := exists_ne (0 : G),
rw ←norm_ne_zero_iff at hg,
have := (of_subsingleton 𝕜 G i').ratio_le_op_norm (λ _, g),
rwa [fintype.prod_subsingleton _ i', of_subsingleton_apply, div_self hg] at this,
end | lemma | continuous_multilinear_map.norm_of_subsingleton | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"div_self",
"exists_ne",
"fintype.prod_subsingleton",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_of_subsingleton_le [subsingleton ι] (i' : ι) : ‖of_subsingleton 𝕜 G i'‖₊ ≤ 1 | norm_of_subsingleton_le _ _ _ | lemma | continuous_multilinear_map.nnnorm_of_subsingleton_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_of_subsingleton [subsingleton ι] [nontrivial G] (i' : ι) :
‖of_subsingleton 𝕜 G i'‖₊ = 1 | nnreal.eq $ norm_of_subsingleton _ _ _ | lemma | continuous_multilinear_map.nnnorm_of_subsingleton | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"nnreal.eq",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_const_of_is_empty [is_empty ι] (x : G) : ‖const_of_is_empty 𝕜 E x‖ = ‖x‖ | begin
apply le_antisymm,
{ refine op_norm_le_bound _ (norm_nonneg _) (λ x, _),
rw [fintype.prod_empty, mul_one, const_of_is_empty_apply], },
{ simpa using (const_of_is_empty 𝕜 E x).le_op_norm 0 }
end | lemma | continuous_multilinear_map.norm_const_of_is_empty | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"fintype.prod_empty",
"is_empty",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnnorm_const_of_is_empty [is_empty ι] (x : G) : ‖const_of_is_empty 𝕜 E x‖₊ = ‖x‖₊ | nnreal.eq $ norm_const_of_is_empty _ _ _ | lemma | continuous_multilinear_map.nnnorm_const_of_is_empty | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"is_empty",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prodL :
(continuous_multilinear_map 𝕜 E G) × (continuous_multilinear_map 𝕜 E G') ≃ₗᵢ[𝕜]
continuous_multilinear_map 𝕜 E (G × G') | { to_fun := λ f, f.1.prod f.2,
inv_fun := λ f, ((continuous_linear_map.fst 𝕜 G G').comp_continuous_multilinear_map f,
(continuous_linear_map.snd 𝕜 G G').comp_continuous_multilinear_map f),
map_add' := λ f g, rfl,
map_smul' := λ c f, rfl,
left_inv := λ f, by ext; refl,
right_inv := λ f, by ext; refl,
n... | def | continuous_multilinear_map.prodL | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_linear_map.fst",
"continuous_linear_map.snd",
"continuous_multilinear_map",
"inv_fun"
] | `continuous_multilinear_map.prod` as a `linear_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
piₗᵢ {ι' : Type v'} [fintype ι'] {E' : ι' → Type wE'} [Π i', normed_add_comm_group (E' i')]
[Π i', normed_space 𝕜 (E' i')] :
@linear_isometry_equiv 𝕜 𝕜 _ _ (ring_hom.id 𝕜) _ _ _
(Π i', continuous_multilinear_map 𝕜 E (E' i')) (continuous_multilinear_map 𝕜 E (Π i, E' i)) _ _
(@pi.module ι' _ 𝕜 _ _ (λ... | { to_linear_equiv :=
-- note: `pi_linear_equiv` does not unify correctly here, presumably due to issues with dependent
-- typeclass arguments.
{ map_add' := λ f g, rfl,
map_smul' := λ c f, rfl,
.. pi_equiv, },
norm_map' := norm_pi } | def | continuous_multilinear_map.piₗᵢ | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"fintype",
"linear_isometry_equiv",
"normed_add_comm_group",
"normed_space",
"pi.module",
"ring_hom.id"
] | `continuous_multilinear_map.pi` as a `linear_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_restrict_scalars : ‖f.restrict_scalars 𝕜'‖ = ‖f‖ | rfl | lemma | continuous_multilinear_map.norm_restrict_scalars | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalarsₗᵢ :
continuous_multilinear_map 𝕜 E G →ₗᵢ[𝕜'] continuous_multilinear_map 𝕜' E G | { to_fun := restrict_scalars 𝕜',
map_add' := λ m₁ m₂, rfl,
map_smul' := λ c m, rfl,
norm_map' := λ f, rfl } | def | continuous_multilinear_map.restrict_scalarsₗᵢ | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"restrict_scalars"
] | `continuous_multilinear_map.restrict_scalars` as a `linear_isometry`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars_linear :
continuous_multilinear_map 𝕜 E G →L[𝕜'] continuous_multilinear_map 𝕜' E G | (restrict_scalarsₗᵢ 𝕜').to_continuous_linear_map | def | continuous_multilinear_map.restrict_scalars_linear | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map"
] | `continuous_multilinear_map.restrict_scalars` as a `continuous_linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_restrict_scalars :
continuous (restrict_scalars 𝕜' : continuous_multilinear_map 𝕜 E G →
continuous_multilinear_map 𝕜' E G) | (restrict_scalars_linear 𝕜').continuous | lemma | continuous_multilinear_map.continuous_restrict_scalars | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous",
"continuous_multilinear_map",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_image_sub_le' [decidable_eq ι] (m₁ m₂ : Πi, E i) :
‖f m₁ - f m₂‖ ≤
‖f‖ * ∑ i, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ | f.to_multilinear_map.norm_image_sub_le_of_bound' (norm_nonneg _) f.le_op_norm _ _ | lemma | continuous_multilinear_map.norm_image_sub_le' | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [] | The difference `f m₁ - f m₂` is controlled in terms of `‖f‖` and `‖m₁ - m₂‖`, precise version.
For a less precise but more usable version, see `norm_image_sub_le`. The bound reads
`‖f m - f m'‖ ≤
‖f‖ * ‖m 1 - m' 1‖ * max ‖m 2‖ ‖m' 2‖ * max ‖m 3‖ ‖m' 3‖ * ... * max ‖m n‖ ‖m' n‖ + ...`,
where the other terms in the sum... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_image_sub_le (m₁ m₂ : Πi, E i) :
‖f m₁ - f m₂‖ ≤ ‖f‖ * (fintype.card ι) * (max ‖m₁‖ ‖m₂‖) ^ (fintype.card ι - 1) * ‖m₁ - m₂‖ | f.to_multilinear_map.norm_image_sub_le_of_bound (norm_nonneg _) f.le_op_norm _ _ | lemma | continuous_multilinear_map.norm_image_sub_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"fintype.card"
] | The difference `f m₁ - f m₂` is controlled in terms of `‖f‖` and `‖m₁ - m₂‖`, less precise
version. For a more precise but less usable version, see `norm_image_sub_le'`.
The bound is `‖f m - f m'‖ ≤ ‖f‖ * card ι * ‖m - m'‖ * (max ‖m‖ ‖m'‖) ^ (card ι - 1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_eval :
continuous (λ p : continuous_multilinear_map 𝕜 E G × Π i, E i, p.1 p.2) | begin
apply continuous_iff_continuous_at.2 (λp, _),
apply continuous_at_of_locally_lipschitz zero_lt_one
((‖p‖ + 1) * (fintype.card ι) * (‖p‖ + 1) ^ (fintype.card ι - 1) + ∏ i, ‖p.2 i‖)
(λq hq, _),
have : 0 ≤ (max ‖q.2‖ ‖p.2‖), by simp,
have : 0 ≤ ‖p‖ + 1 := zero_le_one.trans ((le_add_iff_nonneg_left 1)... | lemma | continuous_multilinear_map.continuous_eval | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous",
"continuous_at_of_locally_lipschitz",
"continuous_multilinear_map",
"dist_triangle",
"fintype.card",
"max_le_max",
"mul_le_mul",
"nat.cast_nonneg",
"norm_fst_le",
"norm_snd_le",
"pow_le_pow_of_le_left",
"pow_nonneg",
"ring",
"zero_le_one",
"zero_lt_one"
] | Applying a multilinear map to a vector is continuous in both coordinates. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_eval_left (m : Π i, E i) :
continuous (λ p : continuous_multilinear_map 𝕜 E G, p m) | continuous_eval.comp (continuous_id.prod_mk continuous_const) | lemma | continuous_multilinear_map.continuous_eval_left | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous",
"continuous_const",
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_eval
{α : Type*} {p : α → continuous_multilinear_map 𝕜 E G} {q : continuous_multilinear_map 𝕜 E G}
(h : has_sum p q) (m : Π i, E i) : has_sum (λ a, p a m) (q m) | begin
dsimp [has_sum] at h ⊢,
convert ((continuous_eval_left m).tendsto _).comp h,
ext s,
simp
end | lemma | continuous_multilinear_map.has_sum_eval | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eval {α : Type*} {p : α → continuous_multilinear_map 𝕜 E G} (hp : summable p)
(m : Π i, E i) : (∑' a, p a) m = ∑' a, p a m | (has_sum_eval hp.has_sum m).tsum_eq.symm | lemma | continuous_multilinear_map.tsum_eval | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multilinear_map.mk_continuous_norm_le (f : multilinear_map 𝕜 E G) {C : ℝ} (hC : 0 ≤ C)
(H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) : ‖f.mk_continuous C H‖ ≤ C | continuous_multilinear_map.op_norm_le_bound _ hC (λm, H m) | lemma | multilinear_map.mk_continuous_norm_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.op_norm_le_bound",
"multilinear_map"
] | If a continuous multilinear map is constructed from a multilinear map via the constructor
`mk_continuous`, then its norm is bounded by the bound given to the constructor if it is
nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multilinear_map.mk_continuous_norm_le' (f : multilinear_map 𝕜 E G) {C : ℝ}
(H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) : ‖f.mk_continuous C H‖ ≤ max C 0 | continuous_multilinear_map.op_norm_le_bound _ (le_max_right _ _) $
λ m, (H m).trans $ mul_le_mul_of_nonneg_right (le_max_left _ _)
(prod_nonneg $ λ _ _, norm_nonneg _) | lemma | multilinear_map.mk_continuous_norm_le' | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.op_norm_le_bound",
"mul_le_mul_of_nonneg_right",
"multilinear_map"
] | If a continuous multilinear map is constructed from a multilinear map via the constructor
`mk_continuous`, then its norm is bounded by the bound given to the constructor if it is
nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restr {k n : ℕ} (f : (G [×n]→L[𝕜] G' : _)) (s : finset (fin n)) (hk : s.card = k) (z : G) :
G [×k]→L[𝕜] G' | (f.to_multilinear_map.restr s hk z).mk_continuous
(‖f‖ * ‖z‖^(n-k)) $ λ v, multilinear_map.restr_norm_le _ _ _ _ f.le_op_norm _ | def | continuous_multilinear_map.restr | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"finset",
"multilinear_map.restr_norm_le"
] | Given a continuous multilinear map `f` on `n` variables (parameterized by `fin n`) and a subset
`s` of `k` of these variables, one gets a new continuous multilinear map on `fin k` by varying
these variables, and fixing the other ones equal to a given value `z`. It is denoted by
`f.restr s hk z`, where `hk` is a proof t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_restr {k n : ℕ} (f : G [×n]→L[𝕜] G') (s : finset (fin n)) (hk : s.card = k) (z : G) :
‖f.restr s hk z‖ ≤ ‖f‖ * ‖z‖ ^ (n - k) | begin
apply multilinear_map.mk_continuous_norm_le,
exact mul_nonneg (norm_nonneg _) (pow_nonneg (norm_nonneg _) _)
end | lemma | continuous_multilinear_map.norm_restr | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"finset",
"multilinear_map.mk_continuous_norm_le",
"pow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mk_pi_algebra_le [nonempty ι] :
‖continuous_multilinear_map.mk_pi_algebra 𝕜 ι A‖ ≤ 1 | begin
have := λ f, @op_norm_le_bound 𝕜 ι (λ i, A) A _ _ _ _ _ _ f _ zero_le_one,
refine this _ _,
intros m,
simp only [continuous_multilinear_map.mk_pi_algebra_apply, one_mul],
exact norm_prod_le' _ univ_nonempty _,
end | lemma | continuous_multilinear_map.norm_mk_pi_algebra_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.mk_pi_algebra_apply",
"one_mul",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mk_pi_algebra_of_empty [is_empty ι] :
‖continuous_multilinear_map.mk_pi_algebra 𝕜 ι A‖ = ‖(1 : A)‖ | begin
apply le_antisymm,
{ have := λ f, @op_norm_le_bound 𝕜 ι (λ i, A) A _ _ _ _ _ _ f _ (norm_nonneg (1 : A)),
refine this _ _,
simp, },
{ convert ratio_le_op_norm _ (λ _, (1 : A)),
simp [eq_empty_of_is_empty (univ : finset ι)], },
end | lemma | continuous_multilinear_map.norm_mk_pi_algebra_of_empty | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"finset",
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mk_pi_algebra [norm_one_class A] :
‖continuous_multilinear_map.mk_pi_algebra 𝕜 ι A‖ = 1 | begin
casesI is_empty_or_nonempty ι,
{ simp [norm_mk_pi_algebra_of_empty] },
{ refine le_antisymm norm_mk_pi_algebra_le _,
convert ratio_le_op_norm _ (λ _, 1); [skip, apply_instance],
simp },
end | lemma | continuous_multilinear_map.norm_mk_pi_algebra | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"is_empty_or_nonempty",
"norm_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mk_pi_algebra_fin_succ_le :
‖continuous_multilinear_map.mk_pi_algebra_fin 𝕜 n.succ A‖ ≤ 1 | begin
have := λ f, @op_norm_le_bound 𝕜 (fin n.succ) (λ i, A) A _ _ _ _ _ _ f _ zero_le_one,
refine this _ _,
intros m,
simp only [continuous_multilinear_map.mk_pi_algebra_fin_apply, one_mul, list.of_fn_eq_map,
fin.prod_univ_def, multiset.coe_map, multiset.coe_prod],
refine (list.norm_prod_le' _).trans_eq... | lemma | continuous_multilinear_map.norm_mk_pi_algebra_fin_succ_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.mk_pi_algebra_fin_apply",
"fin.prod_univ_def",
"list.fin_range_eq_nil",
"list.map_eq_nil",
"list.norm_prod_le'",
"list.of_fn_eq_map",
"multiset.coe_map",
"multiset.coe_prod",
"one_mul",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mk_pi_algebra_fin_le_of_pos (hn : 0 < n) :
‖continuous_multilinear_map.mk_pi_algebra_fin 𝕜 n A‖ ≤ 1 | begin
obtain ⟨n, rfl⟩ := nat.exists_eq_succ_of_ne_zero hn.ne',
exact norm_mk_pi_algebra_fin_succ_le
end | lemma | continuous_multilinear_map.norm_mk_pi_algebra_fin_le_of_pos | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mk_pi_algebra_fin_zero :
‖continuous_multilinear_map.mk_pi_algebra_fin 𝕜 0 A‖ = ‖(1 : A)‖ | begin
refine le_antisymm _ _,
{ have := λ f, @op_norm_le_bound 𝕜 (fin 0) (λ i, A) A _ _ _ _ _ _ f _ (norm_nonneg (1 : A)),
refine this _ _,
simp, },
{ convert ratio_le_op_norm _ (λ _, (1 : A)),
simp }
end | lemma | continuous_multilinear_map.norm_mk_pi_algebra_fin_zero | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mk_pi_algebra_fin [norm_one_class A] :
‖continuous_multilinear_map.mk_pi_algebra_fin 𝕜 n A‖ = 1 | begin
cases n,
{ simp [norm_mk_pi_algebra_fin_zero] },
{ refine le_antisymm norm_mk_pi_algebra_fin_succ_le _,
convert ratio_le_op_norm _ (λ _, 1); [skip, apply_instance],
simp }
end | lemma | continuous_multilinear_map.norm_mk_pi_algebra_fin | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"norm_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_pi_field (z : G) : continuous_multilinear_map 𝕜 (λ(i : ι), 𝕜) G | multilinear_map.mk_continuous
(multilinear_map.mk_pi_ring 𝕜 ι z) (‖z‖)
(λ m, by simp only [multilinear_map.mk_pi_ring_apply, norm_smul, norm_prod,
mul_comm]) | def | continuous_multilinear_map.mk_pi_field | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"mul_comm",
"multilinear_map.mk_continuous",
"multilinear_map.mk_pi_ring",
"multilinear_map.mk_pi_ring_apply",
"norm_prod",
"norm_smul"
] | The canonical continuous multilinear map on `𝕜^ι`, associating to `m` the product of all the
`m i` (multiplied by a fixed reference element `z` in the target module) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_pi_field_apply (z : G) (m : ι → 𝕜) :
(continuous_multilinear_map.mk_pi_field 𝕜 ι z : (ι → 𝕜) → G) m = (∏ i, m i) • z | rfl | lemma | continuous_multilinear_map.mk_pi_field_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.mk_pi_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_pi_field_apply_one_eq_self (f : continuous_multilinear_map 𝕜 (λ(i : ι), 𝕜) G) :
continuous_multilinear_map.mk_pi_field 𝕜 ι (f (λi, 1)) = f | to_multilinear_map_injective f.to_multilinear_map.mk_pi_ring_apply_one_eq_self | lemma | continuous_multilinear_map.mk_pi_field_apply_one_eq_self | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"continuous_multilinear_map.mk_pi_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mk_pi_field (z : G) : ‖continuous_multilinear_map.mk_pi_field 𝕜 ι z‖ = ‖z‖ | (multilinear_map.mk_continuous_norm_le _ (norm_nonneg z) _).antisymm $
by simpa using (continuous_multilinear_map.mk_pi_field 𝕜 ι z).le_op_norm (λ _, 1) | lemma | continuous_multilinear_map.norm_mk_pi_field | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.mk_pi_field",
"multilinear_map.mk_continuous_norm_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_pi_field_eq_iff {z₁ z₂ : G} :
continuous_multilinear_map.mk_pi_field 𝕜 ι z₁ = continuous_multilinear_map.mk_pi_field 𝕜 ι z₂ ↔
z₁ = z₂ | begin
rw [← to_multilinear_map_injective.eq_iff],
exact multilinear_map.mk_pi_ring_eq_iff
end | lemma | continuous_multilinear_map.mk_pi_field_eq_iff | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.mk_pi_field",
"multilinear_map.mk_pi_ring_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_pi_field_zero :
continuous_multilinear_map.mk_pi_field 𝕜 ι (0 : G) = 0 | by ext; rw [mk_pi_field_apply, smul_zero, continuous_multilinear_map.zero_apply] | lemma | continuous_multilinear_map.mk_pi_field_zero | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.mk_pi_field",
"continuous_multilinear_map.zero_apply",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_pi_field_eq_zero_iff (z : G) :
continuous_multilinear_map.mk_pi_field 𝕜 ι z = 0 ↔ z = 0 | by rw [← mk_pi_field_zero, mk_pi_field_eq_iff] | lemma | continuous_multilinear_map.mk_pi_field_eq_zero_iff | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.mk_pi_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_field_equiv : G ≃ₗᵢ[𝕜] (continuous_multilinear_map 𝕜 (λ(i : ι), 𝕜) G) | { to_fun := λ z, continuous_multilinear_map.mk_pi_field 𝕜 ι z,
inv_fun := λ f, f (λi, 1),
map_add' := λ z z', by { ext m, simp [smul_add] },
map_smul' := λ c z, by { ext m, simp [smul_smul, mul_comm] },
left_inv := λ z, by simp,
right_inv := λ f, f.mk_pi_field_apply_one_eq_self,
norm_map' := norm_mk... | def | continuous_multilinear_map.pi_field_equiv | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"continuous_multilinear_map.mk_pi_field",
"inv_fun",
"mul_comm",
"smul_add",
"smul_smul"
] | Continuous multilinear maps on `𝕜^n` with values in `G` are in bijection with `G`, as such a
continuous multilinear map is completely determined by its value on the constant vector made of
ones. We register this bijection as a linear isometry in
`continuous_multilinear_map.pi_field_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_comp_continuous_multilinear_map_le (g : G →L[𝕜] G')
(f : continuous_multilinear_map 𝕜 E G) :
‖g.comp_continuous_multilinear_map f‖ ≤ ‖g‖ * ‖f‖ | continuous_multilinear_map.op_norm_le_bound _ (mul_nonneg (norm_nonneg _) (norm_nonneg _)) $ λ m,
calc ‖g (f m)‖ ≤ ‖g‖ * (‖f‖ * ∏ i, ‖m i‖) : g.le_op_norm_of_le $ f.le_op_norm _
... = _ : (mul_assoc _ _ _).symm | lemma | continuous_linear_map.norm_comp_continuous_multilinear_map_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"continuous_multilinear_map.op_norm_le_bound",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_multilinear_mapL :
(G →L[𝕜] G') →L[𝕜] continuous_multilinear_map 𝕜 E G →L[𝕜] continuous_multilinear_map 𝕜 E G' | linear_map.mk_continuous₂
(linear_map.mk₂ 𝕜 comp_continuous_multilinear_map (λ f₁ f₂ g, rfl) (λ c f g, rfl)
(λ f g₁ g₂, by { ext1, apply f.map_add }) (λ c f g, by { ext1, simp }))
1 $ λ f g, by { rw one_mul, exact f.norm_comp_continuous_multilinear_map_le g } | def | continuous_linear_map.comp_continuous_multilinear_mapL | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"linear_map.mk_continuous₂",
"linear_map.mk₂",
"one_mul"
] | `continuous_linear_map.comp_continuous_multilinear_map` as a bundled continuous bilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.continuous_linear_equiv.comp_continuous_multilinear_mapL (g : G ≃L[𝕜] G') :
continuous_multilinear_map 𝕜 E G ≃L[𝕜] continuous_multilinear_map 𝕜 E G' | { inv_fun := comp_continuous_multilinear_mapL 𝕜 _ _ _ g.symm.to_continuous_linear_map,
left_inv := begin
assume f,
ext1 m,
simp only [comp_continuous_multilinear_mapL, continuous_linear_equiv.coe_def_rev,
to_linear_map_eq_coe, linear_map.to_fun_eq_coe, coe_coe, linear_map.mk_continuous₂_apply,
... | def | continuous_linear_equiv.comp_continuous_multilinear_mapL | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"coe_coe",
"continuous",
"continuous_linear_equiv.apply_symm_apply",
"continuous_linear_equiv.coe_coe",
"continuous_linear_equiv.coe_def_rev",
"continuous_linear_equiv.symm_apply_apply",
"continuous_multilinear_map",
"inv_fun",
"linear_map.mk_continuous₂_apply",
"linear_map.mk₂_apply",
"linear_m... | `continuous_linear_map.comp_continuous_multilinear_map` as a bundled
continuous linear equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.continuous_linear_equiv.comp_continuous_multilinear_mapL_symm
(g : G ≃L[𝕜] G') :
(g.comp_continuous_multilinear_mapL E).symm = g.symm.comp_continuous_multilinear_mapL E | rfl | lemma | continuous_linear_equiv.comp_continuous_multilinear_mapL_symm | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_linear_equiv.comp_continuous_multilinear_mapL_apply
(g : G ≃L[𝕜] G') (f : continuous_multilinear_map 𝕜 E G) :
g.comp_continuous_multilinear_mapL E f = (g : G →L[𝕜] G').comp_continuous_multilinear_map f | rfl | lemma | continuous_linear_equiv.comp_continuous_multilinear_mapL_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip_multilinear (f : G →L[𝕜] continuous_multilinear_map 𝕜 E G') :
continuous_multilinear_map 𝕜 E (G →L[𝕜] G') | multilinear_map.mk_continuous
{ to_fun := λ m, linear_map.mk_continuous
{ to_fun := λ x, f x m,
map_add' := λ x y, by simp only [map_add, continuous_multilinear_map.add_apply],
map_smul' := λ c x, by simp only [continuous_multilinear_map.smul_apply, map_smul,
... | def | continuous_linear_map.flip_multilinear | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"continuous_multilinear_map.add_apply",
"continuous_multilinear_map.map_add",
"continuous_multilinear_map.map_smul",
"continuous_multilinear_map.smul_apply",
"linear_map.coe_mk",
"linear_map.mk_continuous",
"linear_map.mk_continuous_apply",
"linear_map.mk_continuous_nor... | Flip arguments in `f : G →L[𝕜] continuous_multilinear_map 𝕜 E G'` to get
`continuous_multilinear_map 𝕜 E (G →L[𝕜] G')` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry.norm_comp_continuous_multilinear_map
(g : G →ₗᵢ[𝕜] G') (f : continuous_multilinear_map 𝕜 E G) :
‖g.to_continuous_linear_map.comp_continuous_multilinear_map f‖ = ‖f‖ | by simp only [continuous_linear_map.comp_continuous_multilinear_map_coe,
linear_isometry.coe_to_continuous_linear_map, linear_isometry.norm_map,
continuous_multilinear_map.norm_def] | lemma | linear_isometry.norm_comp_continuous_multilinear_map | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_linear_map.comp_continuous_multilinear_map_coe",
"continuous_multilinear_map",
"continuous_multilinear_map.norm_def",
"linear_isometry.coe_to_continuous_linear_map",
"linear_isometry.norm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_continuous_linear (f : G →ₗ[𝕜] multilinear_map 𝕜 E G') (C : ℝ)
(H : ∀ x m, ‖f x m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖) :
G →L[𝕜] continuous_multilinear_map 𝕜 E G' | linear_map.mk_continuous
{ to_fun := λ x, (f x).mk_continuous (C * ‖x‖) $ H x,
map_add' := λ x y, by { ext1, simp only [_root_.map_add], refl },
map_smul' := λ c x, by { ext1, simp only [smul_hom_class.map_smul], refl } }
(max C 0) $ λ x, ((f x).mk_continuous_norm_le' _).trans_eq $
by rw [max_mul_of_non... | def | multilinear_map.mk_continuous_linear | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"linear_map.mk_continuous",
"max_mul_of_nonneg",
"multilinear_map",
"zero_mul"
] | Given a map `f : G →ₗ[𝕜] multilinear_map 𝕜 E G'` and an estimate
`H : ∀ x m, ‖f x m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖`, construct a continuous linear
map from `G` to `continuous_multilinear_map 𝕜 E G'`.
In order to lift, e.g., a map `f : (multilinear_map 𝕜 E G) →ₗ[𝕜] multilinear_map 𝕜 E' G'`
to a map `(continuous_multilin... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_continuous_linear_norm_le' (f : G →ₗ[𝕜] multilinear_map 𝕜 E G') (C : ℝ)
(H : ∀ x m, ‖f x m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖) :
‖mk_continuous_linear f C H‖ ≤ max C 0 | begin
dunfold mk_continuous_linear,
exact linear_map.mk_continuous_norm_le _ (le_max_right _ _) _
end | lemma | multilinear_map.mk_continuous_linear_norm_le' | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"linear_map.mk_continuous_norm_le",
"multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_continuous_linear_norm_le (f : G →ₗ[𝕜] multilinear_map 𝕜 E G') {C : ℝ} (hC : 0 ≤ C)
(H : ∀ x m, ‖f x m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖) :
‖mk_continuous_linear f C H‖ ≤ C | (mk_continuous_linear_norm_le' f C H).trans_eq (max_eq_left hC) | lemma | multilinear_map.mk_continuous_linear_norm_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_continuous_multilinear (f : multilinear_map 𝕜 E (multilinear_map 𝕜 E' G)) (C : ℝ)
(H : ∀ m₁ m₂, ‖f m₁ m₂‖ ≤ C * (∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖) :
continuous_multilinear_map 𝕜 E (continuous_multilinear_map 𝕜 E' G) | mk_continuous
{ to_fun := λ m, mk_continuous (f m) (C * ∏ i, ‖m i‖) $ H m,
map_add' := λ _ m i x y, by { ext1, simp },
map_smul' := λ _ m i c x, by { ext1, simp } }
(max C 0) $ λ m, ((f m).mk_continuous_norm_le' _).trans_eq $
by { rw [max_mul_of_nonneg, zero_mul], exact prod_nonneg (λ _ _, norm_nonneg _... | def | multilinear_map.mk_continuous_multilinear | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"max_mul_of_nonneg",
"multilinear_map",
"zero_mul"
] | Given a map `f : multilinear_map 𝕜 E (multilinear_map 𝕜 E' G)` and an estimate
`H : ∀ m m', ‖f m m'‖ ≤ C * ∏ i, ‖m i‖ * ∏ i, ‖m' i‖`, upgrade all `multilinear_map`s in the type to
`continuous_multilinear_map`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_continuous_multilinear_apply (f : multilinear_map 𝕜 E (multilinear_map 𝕜 E' G))
{C : ℝ} (H : ∀ m₁ m₂, ‖f m₁ m₂‖ ≤ C * (∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖) (m : Π i, E i) :
⇑(mk_continuous_multilinear f C H m) = f m | rfl | lemma | multilinear_map.mk_continuous_multilinear_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_continuous_multilinear_norm_le' (f : multilinear_map 𝕜 E (multilinear_map 𝕜 E' G)) (C : ℝ)
(H : ∀ m₁ m₂, ‖f m₁ m₂‖ ≤ C * (∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖) :
‖mk_continuous_multilinear f C H‖ ≤ max C 0 | begin
dunfold mk_continuous_multilinear,
exact mk_continuous_norm_le _ (le_max_right _ _) _
end | lemma | multilinear_map.mk_continuous_multilinear_norm_le' | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_continuous_multilinear_norm_le (f : multilinear_map 𝕜 E (multilinear_map 𝕜 E' G)) {C : ℝ}
(hC : 0 ≤ C) (H : ∀ m₁ m₂, ‖f m₁ m₂‖ ≤ C * (∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖) :
‖mk_continuous_multilinear f C H‖ ≤ C | (mk_continuous_multilinear_norm_le' f C H).trans_eq (max_eq_left hC) | lemma | multilinear_map.mk_continuous_multilinear_norm_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_comp_continuous_linear_le (g : continuous_multilinear_map 𝕜 E₁ G)
(f : Π i, E i →L[𝕜] E₁ i) :
‖g.comp_continuous_linear_map f‖ ≤ ‖g‖ * ∏ i, ‖f i‖ | op_norm_le_bound _ (mul_nonneg (norm_nonneg _) $ prod_nonneg $ λ i hi, norm_nonneg _) $ λ m,
calc ‖g (λ i, f i (m i))‖ ≤ ‖g‖ * ∏ i, ‖f i (m i)‖ : g.le_op_norm _
... ≤ ‖g‖ * ∏ i, (‖f i‖ * ‖m i‖) :
mul_le_mul_of_nonneg_left
(prod_le_prod (λ _ _, norm_nonneg _) (λ i hi, (f i).le_op_norm (m i))) (norm_nonneg g)
... =... | lemma | continuous_multilinear_map.norm_comp_continuous_linear_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"mul_assoc",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_comp_continuous_linear_isometry_le (g : continuous_multilinear_map 𝕜 E₁ G)
(f : Π i, E i →ₗᵢ[𝕜] E₁ i) :
‖g.comp_continuous_linear_map (λ i, (f i).to_continuous_linear_map)‖ ≤ ‖g‖ | begin
apply op_norm_le_bound _ (norm_nonneg _) (λ m, _),
apply (g.le_op_norm _).trans _,
simp only [continuous_linear_map.to_linear_map_eq_coe, continuous_linear_map.coe_coe,
linear_isometry.coe_to_continuous_linear_map, linear_isometry.norm_map]
end | lemma | continuous_multilinear_map.norm_comp_continuous_linear_isometry_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_linear_map.coe_coe",
"continuous_linear_map.to_linear_map_eq_coe",
"continuous_multilinear_map",
"linear_isometry.coe_to_continuous_linear_map",
"linear_isometry.norm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_comp_continuous_linear_isometry_equiv (g : continuous_multilinear_map 𝕜 E₁ G)
(f : Π i, E i ≃ₗᵢ[𝕜] E₁ i) :
‖g.comp_continuous_linear_map (λ i, (f i : E i →L[𝕜] E₁ i))‖ = ‖g‖ | begin
apply le_antisymm (g.norm_comp_continuous_linear_isometry_le (λ i, (f i).to_linear_isometry)),
have : g = (g.comp_continuous_linear_map (λ i, (f i : E i →L[𝕜] E₁ i)))
.comp_continuous_linear_map (λ i, ((f i).symm : E₁ i →L[𝕜] E i)),
{ ext1 m,
simp only [comp_continuous_linear_map_apply, linear_iso... | lemma | continuous_multilinear_map.norm_comp_continuous_linear_isometry_equiv | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"linear_isometry_equiv.apply_symm_apply",
"linear_isometry_equiv.coe_coe''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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