statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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comp_continuous_linear_mapL (f : Π i, E i →L[𝕜] E₁ i) :
continuous_multilinear_map 𝕜 E₁ G →L[𝕜] continuous_multilinear_map 𝕜 E G | linear_map.mk_continuous
{ to_fun := λ g, g.comp_continuous_linear_map f,
map_add' := λ g₁ g₂, rfl,
map_smul' := λ c g, rfl }
(∏ i, ‖f i‖) $ λ g, (norm_comp_continuous_linear_le _ _).trans_eq (mul_comm _ _) | def | continuous_multilinear_map.comp_continuous_linear_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_continuous",
"mul_comm"
] | `continuous_multilinear_map.comp_continuous_linear_map` as a bundled continuous linear map.
This implementation fixes `f : Π i, E i →L[𝕜] E₁ i`.
TODO: Actually, the map is multilinear in `f` but an attempt to formalize this failed because of
issues with class instances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_linear_mapL_apply
(g : continuous_multilinear_map 𝕜 E₁ G) (f : Π i, E i →L[𝕜] E₁ i) :
comp_continuous_linear_mapL f g = g.comp_continuous_linear_map f | rfl | lemma | continuous_multilinear_map.comp_continuous_linear_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 | |
norm_comp_continuous_linear_mapL_le (f : Π i, E i →L[𝕜] E₁ i) :
‖@comp_continuous_linear_mapL 𝕜 ι E E₁ G _ _ _ _ _ _ _ _ f‖ ≤ (∏ i, ‖f i‖) | linear_map.mk_continuous_norm_le _ (prod_nonneg $ λ i _, norm_nonneg _) _ | lemma | continuous_multilinear_map.norm_comp_continuous_linear_mapL_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_linear_map_equivL (f : Π i, E i ≃L[𝕜] E₁ i) :
continuous_multilinear_map 𝕜 E₁ G ≃L[𝕜] continuous_multilinear_map 𝕜 E G | { inv_fun := comp_continuous_linear_mapL (λ i, ((f i).symm : E₁ i →L[𝕜] E i)),
continuous_to_fun := (comp_continuous_linear_mapL (λ i, (f i : E i →L[𝕜] E₁ i))).continuous,
continuous_inv_fun :=
(comp_continuous_linear_mapL (λ i, ((f i).symm : E₁ i →L[𝕜] E i))).continuous,
left_inv := begin
assume g,
... | def | continuous_multilinear_map.comp_continuous_linear_map_equivL | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous",
"continuous_linear_equiv.apply_symm_apply",
"continuous_linear_equiv.coe_coe",
"continuous_linear_equiv.symm_apply_apply",
"continuous_linear_map.coe_coe",
"continuous_linear_map.to_linear_map_eq_coe",
"continuous_multilinear_map",
"inv_fun",
"linear_map.to_fun_eq_coe"
] | `continuous_multilinear_map.comp_continuous_linear_map` as a bundled continuous linear equiv,
given `f : Π i, E i ≃L[𝕜] E₁ i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_linear_map_equivL_symm (f : Π i, E i ≃L[𝕜] E₁ i) :
(comp_continuous_linear_map_equivL G f).symm =
comp_continuous_linear_map_equivL G (λ (i : ι), (f i).symm) | rfl | lemma | continuous_multilinear_map.comp_continuous_linear_map_equivL_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 | |
comp_continuous_linear_map_equivL_apply
(g : continuous_multilinear_map 𝕜 E₁ G) (f : Π i, E i ≃L[𝕜] E₁ i) :
comp_continuous_linear_map_equivL G f g =
g.comp_continuous_linear_map (λ i, (f i : E i →L[𝕜] E₁ i)) | rfl | lemma | continuous_multilinear_map.comp_continuous_linear_map_equivL_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 | |
continuous_linear_map.norm_map_tail_le
(f : Ei 0 →L[𝕜] (continuous_multilinear_map 𝕜 (λ(i : fin n), Ei i.succ) G)) (m : Πi, Ei i) :
‖f (m 0) (tail m)‖ ≤ ‖f‖ * ∏ i, ‖m i‖ | calc
‖f (m 0) (tail m)‖ ≤ ‖f (m 0)‖ * ∏ i, ‖(tail m) i‖ : (f (m 0)).le_op_norm _
... ≤ (‖f‖ * ‖m 0‖) * ∏ i, ‖(tail m) i‖ :
mul_le_mul_of_nonneg_right (f.le_op_norm _) (prod_nonneg (λi hi, norm_nonneg _))
... = ‖f‖ * (‖m 0‖ * ∏ i, ‖(tail m) i‖) : by ring
... = ‖f‖ * ∏ i, ‖m i‖ : by { rw prod_univ_succ, refl ... | lemma | continuous_linear_map.norm_map_tail_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"mul_le_mul_of_nonneg_right",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.norm_map_init_le
(f : continuous_multilinear_map 𝕜 (λ(i : fin n), Ei i.cast_succ) (Ei (last n) →L[𝕜] G))
(m : Πi, Ei i) :
‖f (init m) (m (last n))‖ ≤ ‖f‖ * ∏ i, ‖m i‖ | calc
‖f (init m) (m (last n))‖ ≤ ‖f (init m)‖ * ‖m (last n)‖ : (f (init m)).le_op_norm _
... ≤ (‖f‖ * (∏ i, ‖(init m) i‖)) * ‖m (last n)‖ :
mul_le_mul_of_nonneg_right (f.le_op_norm _) (norm_nonneg _)
... = ‖f‖ * ((∏ i, ‖(init m) i‖) * ‖m (last n)‖) : mul_assoc _ _ _
... = ‖f‖ * ∏ i, ‖m i‖ : by { rw prod_uni... | lemma | continuous_multilinear_map.norm_map_init_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_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.norm_map_cons_le
(f : continuous_multilinear_map 𝕜 Ei G) (x : Ei 0) (m : Π(i : fin n), Ei i.succ) :
‖f (cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i, ‖m i‖ | calc
‖f (cons x m)‖ ≤ ‖f‖ * ∏ i, ‖cons x m i‖ : f.le_op_norm _
... = (‖f‖ * ‖x‖) * ∏ i, ‖m i‖ : by { rw prod_univ_succ, simp [mul_assoc] } | lemma | continuous_multilinear_map.norm_map_cons_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.norm_map_snoc_le
(f : continuous_multilinear_map 𝕜 Ei G) (m : Π(i : fin n), Ei i.cast_succ) (x : Ei (last n)) :
‖f (snoc m x)‖ ≤ ‖f‖ * (∏ i, ‖m i‖) * ‖x‖ | calc
‖f (snoc m x)‖ ≤ ‖f‖ * ∏ i, ‖snoc m x i‖ : f.le_op_norm _
... = ‖f‖ * (∏ i, ‖m i‖) * ‖x‖ : by { rw prod_univ_cast_succ, simp [mul_assoc] } | lemma | continuous_multilinear_map.norm_map_snoc_le | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.uncurry_left
(f : Ei 0 →L[𝕜] (continuous_multilinear_map 𝕜 (λ(i : fin n), Ei i.succ) G)) :
continuous_multilinear_map 𝕜 Ei G | (@linear_map.uncurry_left 𝕜 n Ei G _ _ _ _ _
(continuous_multilinear_map.to_multilinear_map_linear.comp f.to_linear_map)).mk_continuous
(‖f‖) (λm, continuous_linear_map.norm_map_tail_le f m) | def | continuous_linear_map.uncurry_left | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_linear_map.norm_map_tail_le",
"continuous_multilinear_map",
"linear_map.uncurry_left"
] | Given a continuous linear map `f` from `E 0` to continuous multilinear maps on `n` variables,
construct the corresponding continuous multilinear map on `n+1` variables obtained by concatenating
the variables, given by `m ↦ f (m 0) (tail m)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.uncurry_left_apply
(f : Ei 0 →L[𝕜] (continuous_multilinear_map 𝕜 (λ(i : fin n), Ei i.succ) G)) (m : Πi, Ei i) :
f.uncurry_left m = f (m 0) (tail m) | rfl | lemma | continuous_linear_map.uncurry_left_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 | |
continuous_multilinear_map.curry_left
(f : continuous_multilinear_map 𝕜 Ei G) :
Ei 0 →L[𝕜] (continuous_multilinear_map 𝕜 (λ(i : fin n), Ei i.succ) G) | linear_map.mk_continuous
{ -- define a linear map into `n` continuous multilinear maps from an `n+1` continuous multilinear
-- map
to_fun := λx, (f.to_multilinear_map.curry_left x).mk_continuous
(‖f‖ * ‖x‖) (f.norm_map_cons_le x),
map_add' := λx y, by { ext m, exact f.cons_add m x y },
map_smul' := λc x... | def | continuous_multilinear_map.curry_left | 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",
"multilinear_map.mk_continuous_norm_le"
] | Given a continuous multilinear map `f` in `n+1` variables, split the first variable to obtain
a continuous linear map into continuous multilinear maps in `n` variables, given by
`x ↦ (m ↦ f (cons x m))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_multilinear_map.curry_left_apply
(f : continuous_multilinear_map 𝕜 Ei G) (x : Ei 0) (m : Π(i : fin n), Ei i.succ) :
f.curry_left x m = f (cons x m) | rfl | lemma | continuous_multilinear_map.curry_left_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 | |
continuous_linear_map.curry_uncurry_left
(f : Ei 0 →L[𝕜] (continuous_multilinear_map 𝕜 (λ(i : fin n), Ei i.succ) G)) :
f.uncurry_left.curry_left = f | begin
ext m x,
simp only [tail_cons, continuous_linear_map.uncurry_left_apply,
continuous_multilinear_map.curry_left_apply],
rw cons_zero
end | lemma | continuous_linear_map.curry_uncurry_left | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_linear_map.uncurry_left_apply",
"continuous_multilinear_map",
"continuous_multilinear_map.curry_left_apply",
"tail_cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.uncurry_curry_left
(f : continuous_multilinear_map 𝕜 Ei G) : f.curry_left.uncurry_left = f | continuous_multilinear_map.to_multilinear_map_injective $ f.to_multilinear_map.uncurry_curry_left | lemma | continuous_multilinear_map.uncurry_curry_left | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"continuous_multilinear_map.to_multilinear_map_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_curry_left_equiv :
(Ei 0 →L[𝕜] (continuous_multilinear_map 𝕜 (λ(i : fin n), Ei i.succ) G)) ≃ₗᵢ[𝕜]
(continuous_multilinear_map 𝕜 Ei G) | linear_isometry_equiv.of_bounds
{ to_fun := continuous_linear_map.uncurry_left,
map_add' := λf₁ f₂, by { ext m, refl },
map_smul' := λc f, by { ext m, refl },
inv_fun := continuous_multilinear_map.curry_left,
left_inv := continuous_linear_map.curry_uncurry_left,
right_inv := continuous_mult... | def | continuous_multilinear_curry_left_equiv | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_linear_map.curry_uncurry_left",
"continuous_linear_map.uncurry_left",
"continuous_multilinear_map",
"continuous_multilinear_map.curry_left",
"continuous_multilinear_map.uncurry_curry_left",
"inv_fun",
"linear_isometry_equiv.of_bounds",
"linear_map.mk_continuous_norm_le",
"multilinear_map... | The space of continuous multilinear maps on `Π(i : fin (n+1)), E i` is canonically isomorphic to
the space of continuous linear maps from `E 0` to the space of continuous multilinear maps on
`Π(i : fin n), E i.succ `, by separating the first variable. We register this isomorphism in
`continuous_multilinear_curry_left_e... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_multilinear_curry_left_equiv_apply
(f : Ei 0 →L[𝕜] (continuous_multilinear_map 𝕜 (λ i : fin n, Ei i.succ) G)) (v : Π i, Ei i) :
continuous_multilinear_curry_left_equiv 𝕜 Ei G f v = f (v 0) (tail v) | rfl | lemma | continuous_multilinear_curry_left_equiv_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_left_equiv",
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_curry_left_equiv_symm_apply
(f : continuous_multilinear_map 𝕜 Ei G) (x : Ei 0) (v : Π i : fin n, Ei i.succ) :
(continuous_multilinear_curry_left_equiv 𝕜 Ei G).symm f x v = f (cons x v) | rfl | lemma | continuous_multilinear_curry_left_equiv_symm_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_left_equiv",
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.curry_left_norm
(f : continuous_multilinear_map 𝕜 Ei G) : ‖f.curry_left‖ = ‖f‖ | (continuous_multilinear_curry_left_equiv 𝕜 Ei G).symm.norm_map f | lemma | continuous_multilinear_map.curry_left_norm | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_left_equiv",
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.uncurry_left_norm
(f : Ei 0 →L[𝕜] (continuous_multilinear_map 𝕜 (λ(i : fin n), Ei i.succ) G)) :
‖f.uncurry_left‖ = ‖f‖ | (continuous_multilinear_curry_left_equiv 𝕜 Ei G).norm_map f | lemma | continuous_linear_map.uncurry_left_norm | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_left_equiv",
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.uncurry_right
(f : continuous_multilinear_map 𝕜 (λ i : fin n, Ei i.cast_succ) (Ei (last n) →L[𝕜] G)) :
continuous_multilinear_map 𝕜 Ei G | let f' : multilinear_map 𝕜 (λ(i : fin n), Ei i.cast_succ) (Ei (last n) →ₗ[𝕜] G) :=
{ to_fun := λ m, (f m).to_linear_map,
map_add' := λ _ m i x y, by simp,
map_smul' := λ _ m i c x, by simp } in
(@multilinear_map.uncurry_right 𝕜 n Ei G _ _ _ _ _ f').mk_continuous
(‖f‖) (λm, f.norm_map_init_le m) | def | continuous_multilinear_map.uncurry_right | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"multilinear_map",
"multilinear_map.uncurry_right"
] | Given a continuous linear map `f` from continuous multilinear maps on `n` variables to
continuous linear maps on `E 0`, construct the corresponding continuous multilinear map on `n+1`
variables obtained by concatenating the variables, given by `m ↦ f (init m) (m (last n))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_multilinear_map.uncurry_right_apply
(f : continuous_multilinear_map 𝕜 (λ(i : fin n), Ei i.cast_succ) (Ei (last n) →L[𝕜] G))
(m : Πi, Ei i) :
f.uncurry_right m = f (init m) (m (last n)) | rfl | lemma | continuous_multilinear_map.uncurry_right_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 | |
continuous_multilinear_map.curry_right
(f : continuous_multilinear_map 𝕜 Ei G) :
continuous_multilinear_map 𝕜 (λ i : fin n, Ei i.cast_succ) (Ei (last n) →L[𝕜] G) | let f' : multilinear_map 𝕜 (λ(i : fin n), Ei i.cast_succ) (Ei (last n) →L[𝕜] G) :=
{ to_fun := λm, (f.to_multilinear_map.curry_right m).mk_continuous
(‖f‖ * ∏ i, ‖m i‖) $ λx, f.norm_map_snoc_le m x,
map_add' := λ _ m i x y, by { simp, refl },
map_smul' := λ _ m i c x, by { simp, refl } } in
f'.mk_continuo... | def | continuous_multilinear_map.curry_right | 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_norm_le",
"multilinear_map"
] | Given a continuous multilinear map `f` in `n+1` variables, split the last variable to obtain
a continuous multilinear map in `n` variables into continuous linear maps, given by
`m ↦ (x ↦ f (snoc m x))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_multilinear_map.curry_right_apply
(f : continuous_multilinear_map 𝕜 Ei G) (m : Π i : fin n, Ei i.cast_succ) (x : Ei (last n)) :
f.curry_right m x = f (snoc m x) | rfl | lemma | continuous_multilinear_map.curry_right_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 | |
continuous_multilinear_map.curry_uncurry_right
(f : continuous_multilinear_map 𝕜 (λ i : fin n, Ei i.cast_succ) (Ei (last n) →L[𝕜] G)) :
f.uncurry_right.curry_right = f | begin
ext m x,
simp only [snoc_last, continuous_multilinear_map.curry_right_apply,
continuous_multilinear_map.uncurry_right_apply],
rw init_snoc
end | lemma | continuous_multilinear_map.curry_uncurry_right | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"continuous_multilinear_map.curry_right_apply",
"continuous_multilinear_map.uncurry_right_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.uncurry_curry_right
(f : continuous_multilinear_map 𝕜 Ei G) : f.curry_right.uncurry_right = f | by { ext m, simp } | lemma | continuous_multilinear_map.uncurry_curry_right | 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 | |
continuous_multilinear_curry_right_equiv :
(continuous_multilinear_map 𝕜 (λ(i : fin n), Ei i.cast_succ) (Ei (last n) →L[𝕜] G)) ≃ₗᵢ[𝕜]
(continuous_multilinear_map 𝕜 Ei G) | linear_isometry_equiv.of_bounds
{ to_fun := continuous_multilinear_map.uncurry_right,
map_add' := λf₁ f₂, by { ext m, refl },
map_smul' := λc f, by { ext m, refl },
inv_fun := continuous_multilinear_map.curry_right,
left_inv := continuous_multilinear_map.curry_uncurry_right,
right_inv := co... | def | continuous_multilinear_curry_right_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.curry_right",
"continuous_multilinear_map.curry_uncurry_right",
"continuous_multilinear_map.uncurry_curry_right",
"continuous_multilinear_map.uncurry_right",
"inv_fun",
"linear_isometry_equiv.of_bounds",
"multilinear_map.mk_continuous_norm_le"
... | The space of continuous multilinear maps on `Π(i : fin (n+1)), Ei i` is canonically isomorphic to
the space of continuous multilinear maps on `Π(i : fin n), Ei i.cast_succ` with values in the space
of continuous linear maps on `Ei (last n)`, by separating the last variable. We register this
isomorphism as a continuous ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_multilinear_curry_right_equiv' :
(G [×n]→L[𝕜] (G →L[𝕜] G')) ≃ₗᵢ[𝕜] (G [×n.succ]→L[𝕜] G') | continuous_multilinear_curry_right_equiv 𝕜 (λ (i : fin n.succ), G) G' | def | continuous_multilinear_curry_right_equiv' | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_right_equiv"
] | The space of continuous multilinear maps on `Π(i : fin (n+1)), G` is canonically isomorphic to
the space of continuous multilinear maps on `Π(i : fin n), G` with values in the space
of continuous linear maps on `G`, by separating the last variable. We register this
isomorphism as a continuous linear equiv in `continuou... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_multilinear_curry_right_equiv_apply
(f : (continuous_multilinear_map 𝕜 (λ(i : fin n), Ei i.cast_succ) (Ei (last n) →L[𝕜] G)))
(v : Π i, Ei i) :
(continuous_multilinear_curry_right_equiv 𝕜 Ei G) f v = f (init v) (v (last n)) | rfl | lemma | continuous_multilinear_curry_right_equiv_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_right_equiv",
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_curry_right_equiv_symm_apply
(f : continuous_multilinear_map 𝕜 Ei G)
(v : Π (i : fin n), Ei i.cast_succ) (x : Ei (last n)) :
(continuous_multilinear_curry_right_equiv 𝕜 Ei G).symm f v x = f (snoc v x) | rfl | lemma | continuous_multilinear_curry_right_equiv_symm_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_right_equiv",
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_curry_right_equiv_apply'
(f : G [×n]→L[𝕜] (G →L[𝕜] G')) (v : fin (n + 1) → G) :
continuous_multilinear_curry_right_equiv' 𝕜 n G G' f v = f (init v) (v (last n)) | rfl | lemma | continuous_multilinear_curry_right_equiv_apply' | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_right_equiv'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_curry_right_equiv_symm_apply'
(f : G [×n.succ]→L[𝕜] G') (v : fin n → G) (x : G) :
(continuous_multilinear_curry_right_equiv' 𝕜 n G G').symm f v x = f (snoc v x) | rfl | lemma | continuous_multilinear_curry_right_equiv_symm_apply' | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_right_equiv'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.curry_right_norm
(f : continuous_multilinear_map 𝕜 Ei G) : ‖f.curry_right‖ = ‖f‖ | (continuous_multilinear_curry_right_equiv 𝕜 Ei G).symm.norm_map f | lemma | continuous_multilinear_map.curry_right_norm | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_right_equiv",
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.uncurry_right_norm
(f : continuous_multilinear_map 𝕜 (λ i : fin n, Ei i.cast_succ) (Ei (last n) →L[𝕜] G)) :
‖f.uncurry_right‖ = ‖f‖ | (continuous_multilinear_curry_right_equiv 𝕜 Ei G).norm_map f | lemma | continuous_multilinear_map.uncurry_right_norm | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_right_equiv",
"continuous_multilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.uncurry0
(f : continuous_multilinear_map 𝕜 (λ (i : fin 0), G) G') : G' | f 0 | def | continuous_multilinear_map.uncurry0 | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map"
] | Associating to a continuous multilinear map in `0` variables the unique value it takes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_multilinear_map.curry0 (x : G') : G [×0]→L[𝕜] G' | continuous_multilinear_map.const_of_is_empty 𝕜 _ x | def | continuous_multilinear_map.curry0 | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.const_of_is_empty"
] | Associating to an element `x` of a vector space `E₂` the continuous multilinear map in `0`
variables taking the (unique) value `x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_multilinear_map.curry0_apply (x : G') (m : (fin 0) → G) :
continuous_multilinear_map.curry0 𝕜 G x m = x | rfl | lemma | continuous_multilinear_map.curry0_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.curry0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.uncurry0_apply (f : G [×0]→L[𝕜] G') :
f.uncurry0 = f 0 | rfl | lemma | continuous_multilinear_map.uncurry0_apply | 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 | |
continuous_multilinear_map.apply_zero_curry0 (f : G [×0]→L[𝕜] G') {x : fin 0 → G} :
continuous_multilinear_map.curry0 𝕜 G (f x) = f | by { ext m, simp [(subsingleton.elim _ _ : x = m)] } | lemma | continuous_multilinear_map.apply_zero_curry0 | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.curry0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.uncurry0_curry0 (f : G [×0]→L[𝕜] G') :
continuous_multilinear_map.curry0 𝕜 G (f.uncurry0) = f | by simp | lemma | continuous_multilinear_map.uncurry0_curry0 | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.curry0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.curry0_uncurry0 (x : G') :
(continuous_multilinear_map.curry0 𝕜 G x).uncurry0 = x | rfl | lemma | continuous_multilinear_map.curry0_uncurry0 | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.curry0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.curry0_norm (x : G') :
‖continuous_multilinear_map.curry0 𝕜 G x‖ = ‖x‖ | norm_const_of_is_empty _ _ _ | lemma | continuous_multilinear_map.curry0_norm | 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 | |
continuous_multilinear_map.fin0_apply_norm (f : G [×0]→L[𝕜] G') {x : fin 0 → G} :
‖f x‖ = ‖f‖ | begin
obtain rfl : x = 0 := subsingleton.elim _ _,
refine le_antisymm (by simpa using f.le_op_norm 0) _,
have : ‖continuous_multilinear_map.curry0 𝕜 G (f.uncurry0)‖ ≤ ‖f.uncurry0‖ :=
continuous_multilinear_map.op_norm_le_bound _ (norm_nonneg _) (λm,
by simp [-continuous_multilinear_map.apply_zero_curry... | lemma | continuous_multilinear_map.fin0_apply_norm | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.apply_zero_curry0",
"continuous_multilinear_map.op_norm_le_bound"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_map.uncurry0_norm (f : G [×0]→L[𝕜] G') : ‖f.uncurry0‖ = ‖f‖ | by simp | lemma | continuous_multilinear_map.uncurry0_norm | 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 | |
continuous_multilinear_curry_fin0 : (G [×0]→L[𝕜] G') ≃ₗᵢ[𝕜] G' | { to_fun := λf, continuous_multilinear_map.uncurry0 f,
inv_fun := λf, continuous_multilinear_map.curry0 𝕜 G f,
map_add' := λf g, rfl,
map_smul' := λc f, rfl,
left_inv := continuous_multilinear_map.uncurry0_curry0,
right_inv := continuous_multilinear_map.curry0_uncurry0 𝕜 G,
norm_map' := continuous_... | def | continuous_multilinear_curry_fin0 | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map.curry0",
"continuous_multilinear_map.curry0_uncurry0",
"continuous_multilinear_map.uncurry0",
"continuous_multilinear_map.uncurry0_curry0",
"continuous_multilinear_map.uncurry0_norm",
"inv_fun"
] | The continuous linear isomorphism between elements of a normed space, and continuous multilinear
maps in `0` variables with values in this normed space.
The direct and inverse maps are `uncurry0` and `curry0`. Use these unless you need the full
framework of linear isometric equivs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_multilinear_curry_fin0_apply (f : G [×0]→L[𝕜] G') :
continuous_multilinear_curry_fin0 𝕜 G G' f = f 0 | rfl | lemma | continuous_multilinear_curry_fin0_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_fin0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_curry_fin0_symm_apply (x : G') (v : (fin 0) → G) :
(continuous_multilinear_curry_fin0 𝕜 G G').symm x v = x | rfl | lemma | continuous_multilinear_curry_fin0_symm_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_fin0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_curry_fin1 : (G [×1]→L[𝕜] G') ≃ₗᵢ[𝕜] (G →L[𝕜] G') | (continuous_multilinear_curry_right_equiv 𝕜 (λ (i : fin 1), G) G').symm.trans
(continuous_multilinear_curry_fin0 𝕜 G (G →L[𝕜] G')) | def | continuous_multilinear_curry_fin1 | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_fin0",
"continuous_multilinear_curry_right_equiv"
] | Continuous multilinear maps from `G^1` to `G'` are isomorphic with continuous linear maps from
`G` to `G'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_multilinear_curry_fin1_apply (f : G [×1]→L[𝕜] G') (x : G) :
continuous_multilinear_curry_fin1 𝕜 G G' f x = f (fin.snoc 0 x) | rfl | lemma | continuous_multilinear_curry_fin1_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_fin1",
"fin.snoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_multilinear_curry_fin1_symm_apply
(f : G →L[𝕜] G') (v : (fin 1) → G) :
(continuous_multilinear_curry_fin1 𝕜 G G').symm f v = f (v 0) | rfl | lemma | continuous_multilinear_curry_fin1_symm_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_curry_fin1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_dom_dom_congr (σ : ι ≃ ι') (f : continuous_multilinear_map 𝕜 (λ _ : ι, G) G') :
‖@dom_dom_congr 𝕜 ι G G' _ _ _ _ _ _ _ ι' σ f‖ = ‖f‖ | by simp only [norm_def, linear_equiv.coe_mk, ← σ.prod_comp,
(σ.arrow_congr (equiv.refl G)).surjective.forall, dom_dom_congr_apply, equiv.arrow_congr_apply,
equiv.coe_refl, comp.left_id, comp_app, equiv.symm_apply_apply, id] | theorem | continuous_multilinear_map.norm_dom_dom_congr | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"equiv.coe_refl",
"equiv.refl",
"equiv.symm_apply_apply",
"linear_equiv.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dom_dom_congrₗᵢ (σ : ι ≃ ι') :
continuous_multilinear_map 𝕜 (λ _ : ι, G) G' ≃ₗᵢ[𝕜]
continuous_multilinear_map 𝕜 (λ _ : ι', G) G' | { map_add' := λ _ _, rfl,
map_smul' := λ _ _, rfl,
norm_map' := norm_dom_dom_congr 𝕜 G G' σ,
.. dom_dom_congr_equiv σ } | def | continuous_multilinear_map.dom_dom_congrₗᵢ | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map"
] | An equivalence of the index set defines a linear isometric equivalence between the spaces
of multilinear maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curry_sum (f : continuous_multilinear_map 𝕜 (λ x : ι ⊕ ι', G) G') :
continuous_multilinear_map 𝕜 (λ x : ι, G) (continuous_multilinear_map 𝕜 (λ x : ι', G) G') | multilinear_map.mk_continuous_multilinear (multilinear_map.curry_sum f.to_multilinear_map) (‖f‖) $
λ m m', by simpa [fintype.prod_sum_type, mul_assoc] using f.le_op_norm (sum.elim m m') | def | continuous_multilinear_map.curry_sum | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"fintype.prod_sum_type",
"mul_assoc",
"multilinear_map.curry_sum",
"multilinear_map.mk_continuous_multilinear",
"sum.elim"
] | A continuous multilinear map with variables indexed by `ι ⊕ ι'` defines a continuous multilinear
map with variables indexed by `ι` taking values in the space of continuous multilinear maps with
variables indexed by `ι'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curry_sum_apply (f : continuous_multilinear_map 𝕜 (λ x : ι ⊕ ι', G) G')
(m : ι → G) (m' : ι' → G) :
f.curry_sum m m' = f (sum.elim m m') | rfl | lemma | continuous_multilinear_map.curry_sum_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"sum.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_sum
(f : continuous_multilinear_map 𝕜 (λ x : ι, G) (continuous_multilinear_map 𝕜 (λ x : ι', G) G')) :
continuous_multilinear_map 𝕜 (λ x : ι ⊕ ι', G) G' | multilinear_map.mk_continuous
(to_multilinear_map_linear.comp_multilinear_map f.to_multilinear_map).uncurry_sum (‖f‖) $ λ m,
by simpa [fintype.prod_sum_type, mul_assoc]
using (f (m ∘ sum.inl)).le_of_op_norm_le (m ∘ sum.inr) (f.le_op_norm _) | def | continuous_multilinear_map.uncurry_sum | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"fintype.prod_sum_type",
"mul_assoc",
"multilinear_map.mk_continuous"
] | A continuous multilinear map with variables indexed by `ι` taking values in the space of
continuous multilinear maps with variables indexed by `ι'` defines a continuous multilinear map with
variables indexed by `ι ⊕ ι'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uncurry_sum_apply
(f : continuous_multilinear_map 𝕜 (λ x : ι, G) (continuous_multilinear_map 𝕜 (λ x : ι', G) G'))
(m : ι ⊕ ι' → G) :
f.uncurry_sum m = f (m ∘ sum.inl) (m ∘ sum.inr) | rfl | lemma | continuous_multilinear_map.uncurry_sum_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 | |
curry_sum_equiv : continuous_multilinear_map 𝕜 (λ x : ι ⊕ ι', G) G' ≃ₗᵢ[𝕜]
continuous_multilinear_map 𝕜 (λ x : ι, G) (continuous_multilinear_map 𝕜 (λ x : ι', G) G') | linear_isometry_equiv.of_bounds
{ to_fun := curry_sum,
inv_fun := uncurry_sum,
map_add' := λ f g, by { ext, refl },
map_smul' := λ c f, by { ext, refl },
left_inv := λ f, by { ext m, exact congr_arg f (sum.elim_comp_inl_inr m) },
right_inv := λ f, by { ext m₁ m₂, change f _ _ = f _ _,
rw [su... | def | continuous_multilinear_map.curry_sum_equiv | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"continuous_multilinear_map",
"inv_fun",
"linear_isometry_equiv.of_bounds",
"multilinear_map.mk_continuous_multilinear_norm_le",
"multilinear_map.mk_continuous_norm_le",
"sum.elim_comp_inl",
"sum.elim_comp_inl_inr",
"sum.elim_comp_inr"
] | Linear isometric equivalence between the space of continuous multilinear maps with variables
indexed by `ι ⊕ ι'` and the space of continuous multilinear maps with variables indexed by `ι`
taking values in the space of continuous multilinear maps with variables indexed by `ι'`.
The forward and inverse functions are `co... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curry_fin_finset {k l n : ℕ} {s : finset (fin n)}
(hk : s.card = k) (hl : sᶜ.card = l) :
(G [×n]→L[𝕜] G') ≃ₗᵢ[𝕜] (G [×k]→L[𝕜] G [×l]→L[𝕜] G') | (dom_dom_congrₗᵢ 𝕜 G G' (fin_sum_equiv_of_finset hk hl).symm).trans
(curry_sum_equiv 𝕜 (fin k) (fin l) G G') | def | continuous_multilinear_map.curry_fin_finset | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"fin_sum_equiv_of_finset",
"finset"
] | If `s : finset (fin n)` is a finite set of cardinality `k` and its complement has cardinality
`l`, then the space of continuous multilinear maps `G [×n]→L[𝕜] G'` of `n` variables is isomorphic
to the space of continuous multilinear maps `G [×k]→L[𝕜] G [×l]→L[𝕜] G'` of `k` variables taking
values in the space of cont... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curry_fin_finset_apply (hk : s.card = k) (hl : sᶜ.card = l)
(f : G [×n]→L[𝕜] G') (mk : fin k → G) (ml : fin l → G) :
curry_fin_finset 𝕜 G G' hk hl f mk ml =
f (λ i, sum.elim mk ml ((fin_sum_equiv_of_finset hk hl).symm i)) | rfl | lemma | continuous_multilinear_map.curry_fin_finset_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"fin_sum_equiv_of_finset",
"sum.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_fin_finset_symm_apply (hk : s.card = k) (hl : sᶜ.card = l)
(f : G [×k]→L[𝕜] G [×l]→L[𝕜] G') (m : fin n → G) :
(curry_fin_finset 𝕜 G G' hk hl).symm f m =
f (λ i, m $ fin_sum_equiv_of_finset hk hl (sum.inl i))
(λ i, m $ fin_sum_equiv_of_finset hk hl (sum.inr i)) | rfl | lemma | continuous_multilinear_map.curry_fin_finset_symm_apply | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"fin_sum_equiv_of_finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_fin_finset_symm_apply_piecewise_const (hk : s.card = k) (hl : sᶜ.card = l)
(f : G [×k]→L[𝕜] G [×l]→L[𝕜] G') (x y : G) :
(curry_fin_finset 𝕜 G G' hk hl).symm f (s.piecewise (λ _, x) (λ _, y)) = f (λ _, x) (λ _, y) | multilinear_map.curry_fin_finset_symm_apply_piecewise_const hk hl _ x y | lemma | continuous_multilinear_map.curry_fin_finset_symm_apply_piecewise_const | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"multilinear_map.curry_fin_finset_symm_apply_piecewise_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_fin_finset_symm_apply_const (hk : s.card = k) (hl : sᶜ.card = l)
(f : G [×k]→L[𝕜] G [×l]→L[𝕜] G') (x : G) :
(curry_fin_finset 𝕜 G G' hk hl).symm f (λ _, x) = f (λ _, x) (λ _, x) | rfl | lemma | continuous_multilinear_map.curry_fin_finset_symm_apply_const | 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 | |
curry_fin_finset_apply_const (hk : s.card = k) (hl : sᶜ.card = l)
(f : G [×n]→L[𝕜] G') (x y : G) :
curry_fin_finset 𝕜 G G' hk hl f (λ _, x) (λ _, y) = f (s.piecewise (λ _, x) (λ _, y)) | begin
refine (curry_fin_finset_symm_apply_piecewise_const hk hl _ _ _).symm.trans _, -- `rw` fails
rw linear_isometry_equiv.symm_apply_apply
end | lemma | continuous_multilinear_map.curry_fin_finset_apply_const | analysis.normed_space | src/analysis/normed_space/multilinear.lean | [
"analysis.normed_space.operator_norm",
"topology.algebra.module.multilinear"
] | [
"linear_isometry_equiv.symm_apply_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Lprojection (P : M) : Prop | (proj : is_idempotent_elem P)
(Lnorm : ∀ (x : X), ‖x‖ = ‖P • x‖ + ‖(1 - P) • x‖) | structure | is_Lprojection | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"is_idempotent_elem"
] | A projection on a normed space `X` is said to be an L-projection if, for all `x` in `X`,
$\|x\| = \|P x\| + \|(1 - P) x\|$.
Note that we write `P • x` instead of `P x` for reasons described in the module docstring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_Mprojection (P : M) : Prop | (proj : is_idempotent_elem P)
(Mnorm : ∀ (x : X), ‖x‖ = (max ‖P • x‖ ‖(1 - P) • x‖)) | structure | is_Mprojection | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"is_idempotent_elem"
] | A projection on a normed space `X` is said to be an M-projection if, for all `x` in `X`,
$\|x\| = max(\|P x\|,\|(1 - P) x\|)$.
Note that we write `P • x` instead of `P x` for reasons described in the module docstring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lcomplement {P : M} (h: is_Lprojection X P) : is_Lprojection X (1 - P) | ⟨h.proj.one_sub, λ x, by { rw [add_comm, sub_sub_cancel], exact h.Lnorm x }⟩ | lemma | is_Lprojection.Lcomplement | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"is_Lprojection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Lcomplement_iff (P : M) : is_Lprojection X P ↔ is_Lprojection X (1 - P) | ⟨Lcomplement, λ h, sub_sub_cancel 1 P ▸ h.Lcomplement⟩ | lemma | is_Lprojection.Lcomplement_iff | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"is_Lprojection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute [has_faithful_smul M X] {P Q : M} (h₁ : is_Lprojection X P)
(h₂ : is_Lprojection X Q) : commute P Q | begin
have PR_eq_RPR : ∀ R : M, is_Lprojection X R → P * R = R * P * R := λ R h₃,
begin
refine @eq_of_smul_eq_smul _ X _ _ _ _ (λ x, _),
rw ← norm_sub_eq_zero_iff,
have e1 : ‖R • x‖ ≥ ‖R • x‖ + 2 • ‖ (P * R) • x - (R * P * R) • x‖ :=
calc ‖R • x‖ = ‖R • (P • (R • x))‖ + ‖(1 - R) • (P • (R • x))‖ +
... | lemma | is_Lprojection.commute | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"commute",
"has_faithful_smul",
"is_Lprojection",
"mul_le_mul_left",
"one_mul",
"one_smul",
"smul_sub",
"sub_smul",
"two_mul",
"two_smul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul [has_faithful_smul M X] {P Q : M} (h₁ : is_Lprojection X P) (h₂ : is_Lprojection X Q) :
is_Lprojection X (P * Q) | begin
refine ⟨is_idempotent_elem.mul_of_commute (h₁.commute h₂) h₁.proj h₂.proj, _⟩,
intro x,
refine le_antisymm _ _,
{ calc ‖ x ‖ = ‖(P * Q) • x + (x - (P * Q) • x)‖ : by rw add_sub_cancel'_right ((P * Q) • x) x
... ≤ ‖(P * Q) • x‖ + ‖ x - (P * Q) • x ‖ : by apply norm_add_le
... = ‖(P * Q) • x‖ + ‖(1 ... | lemma | is_Lprojection.mul | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"has_faithful_smul",
"is_Lprojection",
"one_smul",
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
join [has_faithful_smul M X] {P Q : M} (h₁ : is_Lprojection X P) (h₂ : is_Lprojection X Q) :
is_Lprojection X (P + Q - P * Q) | begin
convert (Lcomplement_iff _).mp (h₁.Lcomplement.mul h₂.Lcomplement) using 1,
noncomm_ring,
end | lemma | is_Lprojection.join | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"has_faithful_smul",
"is_Lprojection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_compl (P : {P : M // is_Lprojection X P}) :
↑(Pᶜ) = (1 : M) - ↑P | rfl | lemma | is_Lprojection.coe_compl | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"is_Lprojection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf [has_faithful_smul M X] (P Q : {P : M // is_Lprojection X P}) :
↑(P ⊓ Q) = ((↑P : (M)) * ↑Q) | rfl | lemma | is_Lprojection.coe_inf | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"has_faithful_smul",
"is_Lprojection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup [has_faithful_smul M X] (P Q : {P : M // is_Lprojection X P}) :
↑(P ⊔ Q) = ((↑P : M) + ↑Q - ↑P * ↑Q) | rfl | lemma | is_Lprojection.coe_sup | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"has_faithful_smul",
"is_Lprojection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sdiff [has_faithful_smul M X] (P Q : {P : M // is_Lprojection X P}) :
↑(P \ Q) = (↑P : M) * (1 - ↑Q) | rfl | lemma | is_Lprojection.coe_sdiff | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"has_faithful_smul",
"is_Lprojection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_def [has_faithful_smul M X] (P Q : {P : M // is_Lprojection X P}) :
P ≤ Q ↔ (P : M) = ↑(P ⊓ Q) | iff.rfl | lemma | is_Lprojection.le_def | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"has_faithful_smul",
"is_Lprojection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ↑(0 : {P : M // is_Lprojection X P}) = (0 : M) | rfl | lemma | is_Lprojection.coe_zero | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"is_Lprojection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ↑(1 : {P : M // is_Lprojection X P}) = (1 : M) | rfl | lemma | is_Lprojection.coe_one | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"is_Lprojection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot [has_faithful_smul M X] :
↑(bounded_order.bot : {P : M // is_Lprojection X P}) = (0 : M) | rfl | lemma | is_Lprojection.coe_bot | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"has_faithful_smul",
"is_Lprojection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top [has_faithful_smul M X] :
↑(bounded_order.top : {P : M // is_Lprojection X P}) = (1 : M) | rfl | lemma | is_Lprojection.coe_top | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"has_faithful_smul",
"is_Lprojection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_mul {P : {P : M // is_Lprojection X P}} {Q : M} :
↑Pᶜ * Q = Q - ↑P * Q | by rw [coe_compl, sub_mul, one_mul] | lemma | is_Lprojection.compl_mul | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"is_Lprojection",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_compl_self {P : {P : M // is_Lprojection X P}} :
(↑P : M) * (↑Pᶜ) = 0 | by rw [coe_compl, mul_sub, mul_one, P.prop.proj.eq, sub_self] | lemma | is_Lprojection.mul_compl_self | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"is_Lprojection",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distrib_lattice_lemma [has_faithful_smul M X] {P Q R : {P : M // is_Lprojection X P}} :
((↑P : M) + ↑Pᶜ * R) * (↑P + ↑Q * ↑R * ↑Pᶜ) = (↑P + ↑Q * ↑R * ↑Pᶜ) | by rw [add_mul, mul_add, mul_add, mul_assoc ↑Pᶜ ↑R (↑Q * ↑R * ↑Pᶜ), ← mul_assoc ↑R (↑Q * ↑R) ↑Pᶜ,
← coe_inf Q, (Pᶜ.prop.commute R.prop).eq, ((Q ⊓ R).prop.commute Pᶜ.prop).eq,
(R.prop.commute (Q ⊓ R).prop).eq, coe_inf Q, mul_assoc ↑Q, ← mul_assoc, mul_assoc ↑R,
(Pᶜ.prop.commute P.prop).eq, mul_compl_self, ze... | lemma | is_Lprojection.distrib_lattice_lemma | analysis.normed_space | src/analysis/normed_space/M_structure.lean | [
"algebra.ring.idempotents",
"tactic.noncomm_ring",
"analysis.normed.group.basic"
] | [
"has_faithful_smul",
"is_Lprojection",
"mul_assoc",
"mul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_image_of_norm_zero [semilinear_map_class 𝓕 σ₁₂ E F] (f : 𝓕)
(hf : continuous f) {x : E} (hx : ‖x‖ = 0) : ‖f x‖ = 0 | begin
refine le_antisymm (le_of_forall_pos_le_add (λ ε hε, _)) (norm_nonneg (f x)),
rcases normed_add_comm_group.tendsto_nhds_nhds.1 (hf.tendsto 0) ε hε with ⟨δ, δ_pos, hδ⟩,
replace hδ := hδ x,
rw [sub_zero, hx] at hδ,
replace hδ := le_of_lt (hδ δ_pos),
rw [map_zero, sub_zero] at hδ,
rwa [zero_add]
end | lemma | norm_image_of_norm_zero | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous",
"semilinear_map_class"
] | If `‖x‖ = 0` and `f` is continuous then `‖f x‖ = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semilinear_map_class.bound_of_shell_semi_normed [semilinear_map_class 𝓕 σ₁₂ E F]
(f : 𝓕) {ε C : ℝ} (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < ‖c‖)
(hf : ∀ x, ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) {x : E} (hx : ‖x‖ ≠ 0) :
‖f x‖ ≤ C * ‖x‖ | begin
rcases rescale_to_shell_semi_normed hc ε_pos hx with ⟨δ, hδ, δxle, leδx, δinv⟩,
have := hf (δ • x) leδx δxle,
simpa only [map_smulₛₗ, norm_smul, mul_left_comm C, mul_le_mul_left (norm_pos_iff.2 hδ),
ring_hom_isometric.is_iso] using hf (δ • x) leδx δxle
end | lemma | semilinear_map_class.bound_of_shell_semi_normed | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"mul_le_mul_left",
"mul_left_comm",
"norm_smul",
"rescale_to_shell_semi_normed",
"semilinear_map_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semilinear_map_class.bound_of_continuous [semilinear_map_class 𝓕 σ₁₂ E F] (f : 𝓕)
(hf : continuous f) : ∃ C, 0 < C ∧ (∀ x : E, ‖f x‖ ≤ C * ‖x‖) | begin
rcases normed_add_comm_group.tendsto_nhds_nhds.1 (hf.tendsto 0) 1 zero_lt_one with ⟨ε, ε_pos, hε⟩,
simp only [sub_zero, map_zero] at hε,
rcases normed_field.exists_one_lt_norm 𝕜 with ⟨c, hc⟩,
have : 0 < ‖c‖ / ε, from div_pos (zero_lt_one.trans hc) ε_pos,
refine ⟨‖c‖ / ε, this, λ x, _⟩,
by_cases hx : ... | lemma | semilinear_map_class.bound_of_continuous | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous",
"div_le_iff'",
"div_pos",
"mul_zero",
"norm_image_of_norm_zero",
"normed_field.exists_one_lt_norm",
"one_div_div",
"semilinear_map_class",
"semilinear_map_class.bound_of_shell_semi_normed",
"zero_lt_one"
] | A continuous linear map between seminormed spaces is bounded when the field is nontrivially
normed. The continuity ensures boundedness on a ball of some radius `ε`. The nontriviality of the
norm is then used to rescale any element into an element of norm in `[ε/C, ε]`, whose image has a
controlled norm. The norm contro... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bound [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) :
∃ C, 0 < C ∧ (∀ x : E, ‖f x‖ ≤ C * ‖x‖) | semilinear_map_class.bound_of_continuous f f.2 | theorem | continuous_linear_map.bound | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"bound",
"ring_hom_isometric",
"semilinear_map_class.bound_of_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.linear_isometry.to_span_singleton {v : E} (hv : ‖v‖ = 1) : 𝕜 →ₗᵢ[𝕜] E | { norm_map' := λ x, by simp [norm_smul, hv],
.. linear_map.to_span_singleton 𝕜 E v } | def | linear_isometry.to_span_singleton | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"linear_map.to_span_singleton",
"norm_smul"
] | Given a unit-length element `x` of a normed space `E` over a field `𝕜`, the natural linear
isometry map from `𝕜` to `E` by taking multiples of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.linear_isometry.to_span_singleton_apply {v : E} (hv : ‖v‖ = 1) (a : 𝕜) :
linear_isometry.to_span_singleton 𝕜 E hv a = a • v | rfl | lemma | linear_isometry.to_span_singleton_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"linear_isometry.to_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.linear_isometry.coe_to_span_singleton {v : E} (hv : ‖v‖ = 1) :
(linear_isometry.to_span_singleton 𝕜 E hv).to_linear_map = linear_map.to_span_singleton 𝕜 E v | rfl | lemma | linear_isometry.coe_to_span_singleton | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"linear_isometry.to_span_singleton",
"linear_map.to_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm (f : E →SL[σ₁₂] F) | Inf {c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖} | def | continuous_linear_map.op_norm | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | The operator norm of a continuous linear map is the inf of all its bounds. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_op_norm : has_norm (E →SL[σ₁₂] F) | ⟨op_norm⟩ | instance | continuous_linear_map.has_op_norm | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"has_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_def (f : E →SL[σ₁₂] F) : ‖f‖ = Inf {c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖} | rfl | lemma | continuous_linear_map.norm_def | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounds_nonempty [ring_hom_isometric σ₁₂] {f : E →SL[σ₁₂] F} :
∃ c, c ∈ { c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖ } | let ⟨M, hMp, hMb⟩ := f.bound in ⟨M, le_of_lt hMp, hMb⟩ | lemma | continuous_linear_map.bounds_nonempty | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"ring_hom_isometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounds_bdd_below {f : E →SL[σ₁₂] F} :
bdd_below { c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖ } | ⟨0, λ _ ⟨hn, _⟩, hn⟩ | lemma | continuous_linear_map.bounds_bdd_below | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_le_bound (f : E →SL[σ₁₂] F) {M : ℝ} (hMp: 0 ≤ M) (hM : ∀ x, ‖f x‖ ≤ M * ‖x‖) :
‖f‖ ≤ M | cInf_le bounds_bdd_below ⟨hMp, hM⟩ | lemma | continuous_linear_map.op_norm_le_bound | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"cInf_le"
] | If one controls the norm of every `A x`, then one controls the norm of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_le_bound' (f : E →SL[σ₁₂] F) {M : ℝ} (hMp: 0 ≤ M)
(hM : ∀ x, ‖x‖ ≠ 0 → ‖f x‖ ≤ M * ‖x‖) :
‖f‖ ≤ M | op_norm_le_bound f hMp $ λ x, (ne_or_eq (‖x‖) 0).elim (hM x) $
λ h, by simp only [h, mul_zero, norm_image_of_norm_zero f f.2 h] | lemma | continuous_linear_map.op_norm_le_bound' | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"mul_zero",
"ne_or_eq",
"norm_image_of_norm_zero"
] | If one controls the norm of every `A x`, `‖x‖ ≠ 0`, then one controls the norm of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_le_of_lipschitz {f : E →SL[σ₁₂] F} {K : ℝ≥0} (hf : lipschitz_with K f) :
‖f‖ ≤ K | f.op_norm_le_bound K.2 $ λ x, by simpa only [dist_zero_right, f.map_zero] using hf.dist_le_mul x 0 | theorem | continuous_linear_map.op_norm_le_of_lipschitz | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_eq_of_bounds {φ : E →SL[σ₁₂] F} {M : ℝ} (M_nonneg : 0 ≤ M)
(h_above : ∀ x, ‖φ x‖ ≤ M*‖x‖) (h_below : ∀ N ≥ 0, (∀ x, ‖φ x‖ ≤ N*‖x‖) → M ≤ N) :
‖φ‖ = M | le_antisymm (φ.op_norm_le_bound M_nonneg h_above)
((le_cInf_iff continuous_linear_map.bounds_bdd_below ⟨M, M_nonneg, h_above⟩).mpr $
λ N ⟨N_nonneg, hN⟩, h_below N N_nonneg hN) | lemma | continuous_linear_map.op_norm_eq_of_bounds | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.bounds_bdd_below",
"le_cInf_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_neg (f : E →SL[σ₁₂] F) : ‖-f‖ = ‖f‖ | by simp only [norm_def, neg_apply, norm_neg] | lemma | continuous_linear_map.op_norm_neg | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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