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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|---|---|---|---|---|---|---|---|---|---|---|
le_op_norm : ‖f x‖ ≤ ‖f‖ * ‖x‖ | begin
obtain ⟨C, Cpos, hC⟩ := f.bound,
replace hC := hC x,
by_cases h : ‖x‖ = 0,
{ rwa [h, mul_zero] at ⊢ hC },
have hlt : 0 < ‖x‖ := lt_of_le_of_ne (norm_nonneg x) (ne.symm h),
exact (div_le_iff hlt).mp (le_cInf bounds_nonempty (λ c ⟨_, hc⟩,
(div_le_iff hlt).mpr $ by { apply hc })),
end | theorem | continuous_linear_map.le_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"
] | [
"div_le_iff",
"le_cInf",
"mul_zero"
] | The fundamental property of the operator norm: `‖f x‖ ≤ ‖f‖ * ‖x‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_op_norm (x y : E) : dist (f x) (f y) ≤ ‖f‖ * dist x y | by simp_rw [dist_eq_norm, ← map_sub, f.le_op_norm] | theorem | continuous_linear_map.dist_le_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_op_norm_le {c : ℝ} (h : ‖f‖ ≤ c) (x : E) : ‖f x‖ ≤ c * ‖x‖ | (f.le_op_norm x).trans (mul_le_mul_of_nonneg_right h (norm_nonneg x)) | theorem | continuous_linear_map.le_of_op_norm_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ratio_le_op_norm : ‖f x‖ / ‖x‖ ≤ ‖f‖ | div_le_of_nonneg_of_le_mul (norm_nonneg _) f.op_norm_nonneg (le_op_norm _ _) | lemma | continuous_linear_map.ratio_le_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"
] | [
"div_le_of_nonneg_of_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_le_op_norm : ‖x‖ ≤ 1 → ‖f x‖ ≤ ‖f‖ | mul_one ‖f‖ ▸ f.le_op_norm_of_le | lemma | continuous_linear_map.unit_le_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"
] | [
"mul_one"
] | The image of the unit ball under a continuous linear map is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_le_of_shell {f : E →SL[σ₁₂] F} {ε C : ℝ} (ε_pos : 0 < ε) (hC : 0 ≤ C)
{c : 𝕜} (hc : 1 < ‖c‖) (hf : ∀ x, ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) :
‖f‖ ≤ C | f.op_norm_le_bound' hC $ λ x hx, semilinear_map_class.bound_of_shell_semi_normed f ε_pos hc hf hx | lemma | continuous_linear_map.op_norm_le_of_shell | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"semilinear_map_class.bound_of_shell_semi_normed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_le_of_ball {f : E →SL[σ₁₂] F} {ε : ℝ} {C : ℝ} (ε_pos : 0 < ε) (hC : 0 ≤ C)
(hf : ∀ x ∈ ball (0 : E) ε, ‖f x‖ ≤ C * ‖x‖) : ‖f‖ ≤ C | begin
rcases normed_field.exists_one_lt_norm 𝕜 with ⟨c, hc⟩,
refine op_norm_le_of_shell ε_pos hC hc (λ x _ hx, hf x _),
rwa ball_zero_eq
end | lemma | continuous_linear_map.op_norm_le_of_ball | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"normed_field.exists_one_lt_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_le_of_nhds_zero {f : E →SL[σ₁₂] F} {C : ℝ} (hC : 0 ≤ C)
(hf : ∀ᶠ x in 𝓝 (0 : E), ‖f x‖ ≤ C * ‖x‖) : ‖f‖ ≤ C | let ⟨ε, ε0, hε⟩ := metric.eventually_nhds_iff_ball.1 hf in op_norm_le_of_ball ε0 hC hε | lemma | continuous_linear_map.op_norm_le_of_nhds_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_le_of_shell' {f : E →SL[σ₁₂] F} {ε C : ℝ} (ε_pos : 0 < ε) (hC : 0 ≤ C)
{c : 𝕜} (hc : ‖c‖ < 1) (hf : ∀ x, ε * ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) :
‖f‖ ≤ C | begin
by_cases h0 : c = 0,
{ refine op_norm_le_of_ball ε_pos hC (λ x hx, hf x _ _),
{ simp [h0] },
{ rwa ball_zero_eq at hx } },
{ rw [← inv_inv c, norm_inv,
inv_lt_one_iff_of_pos (norm_pos_iff.2 $ inv_ne_zero h0)] at hc,
refine op_norm_le_of_shell ε_pos hC hc _,
rwa [norm_inv, div_eq_mul_in... | lemma | continuous_linear_map.op_norm_le_of_shell' | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"div_eq_mul_inv",
"inv_inv",
"inv_lt_one_iff_of_pos",
"inv_ne_zero",
"norm_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_le_of_unit_norm [normed_space ℝ E] [normed_space ℝ F] {f : E →L[ℝ] F} {C : ℝ}
(hC : 0 ≤ C) (hf : ∀ x, ‖x‖ = 1 → ‖f x‖ ≤ C) : ‖f‖ ≤ C | begin
refine op_norm_le_bound' f hC (λ x hx, _),
have H₁ : ‖(‖x‖⁻¹ • x)‖ = 1, by rw [norm_smul, norm_inv, norm_norm, inv_mul_cancel hx],
have H₂ := hf _ H₁,
rwa [map_smul, norm_smul, norm_inv, norm_norm, ← div_eq_inv_mul, div_le_iff] at H₂,
exact (norm_nonneg x).lt_of_ne' hx
end | lemma | continuous_linear_map.op_norm_le_of_unit_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"
] | [
"div_eq_inv_mul",
"div_le_iff",
"inv_mul_cancel",
"norm_inv",
"norm_norm",
"norm_smul",
"normed_space"
] | For a continuous real linear map `f`, if one controls the norm of every `f x`, `‖x‖ = 1`, then
one controls the norm of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_add_le : ‖f + g‖ ≤ ‖f‖ + ‖g‖ | (f + g).op_norm_le_bound (add_nonneg f.op_norm_nonneg g.op_norm_nonneg) $
λ x, (norm_add_le_of_le (f.le_op_norm x) (g.le_op_norm x)).trans_eq (add_mul _ _ _).symm | theorem | continuous_linear_map.op_norm_add_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | The operator norm satisfies the triangle inequality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_zero : ‖(0 : E →SL[σ₁₂] F)‖ = 0 | le_antisymm (cInf_le bounds_bdd_below
⟨le_rfl, λ _, le_of_eq (by { rw [zero_mul], exact norm_zero })⟩)
(op_norm_nonneg _) | theorem | continuous_linear_map.op_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"
] | [
"cInf_le",
"zero_mul"
] | The norm of the `0` operator is `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_id_le : ‖id 𝕜 E‖ ≤ 1 | op_norm_le_bound _ zero_le_one (λx, by simp) | lemma | continuous_linear_map.norm_id_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"zero_le_one"
] | The norm of the identity is at most `1`. It is in fact `1`, except when the space is trivial
where it is `0`. It means that one can not do better than an inequality in general. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_id_of_nontrivial_seminorm (h : ∃ (x : E), ‖x‖ ≠ 0) : ‖id 𝕜 E‖ = 1 | le_antisymm norm_id_le $ let ⟨x, hx⟩ := h in
have _ := (id 𝕜 E).ratio_le_op_norm x,
by rwa [id_apply, div_self hx] at this | lemma | continuous_linear_map.norm_id_of_nontrivial_seminorm | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"div_self"
] | If there is an element with norm different from `0`, then the norm of the identity equals `1`.
(Since we are working with seminorms supposing that the space is non-trivial is not enough.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_smul_le {𝕜' : Type*} [normed_field 𝕜'] [normed_space 𝕜' F]
[smul_comm_class 𝕜₂ 𝕜' F] (c : 𝕜') (f : E →SL[σ₁₂] F) : ‖c • f‖ ≤ ‖c‖ * ‖f‖ | ((c • f).op_norm_le_bound
(mul_nonneg (norm_nonneg _) (op_norm_nonneg _)) (λ _,
begin
erw [norm_smul, mul_assoc],
exact mul_le_mul_of_nonneg_left (le_op_norm _ _) (norm_nonneg _)
end)) | lemma | continuous_linear_map.op_norm_smul_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"mul_assoc",
"mul_le_mul_of_nonneg_left",
"norm_smul",
"normed_field",
"normed_space",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tmp_seminormed_add_comm_group : seminormed_add_comm_group (E →SL[σ₁₂] F) | add_group_seminorm.to_seminormed_add_comm_group
{ to_fun := norm,
map_zero' := op_norm_zero,
add_le' := op_norm_add_le,
neg' := op_norm_neg } | def | continuous_linear_map.tmp_seminormed_add_comm_group | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"seminormed_add_comm_group"
] | Continuous linear maps themselves form a seminormed space with respect to
the operator norm. This is only a temporary definition because we want to replace the topology
with `continuous_linear_map.topological_space` to avoid diamond issues.
See Note [forgetful inheritance] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tmp_pseudo_metric_space : pseudo_metric_space (E →SL[σ₁₂] F) | continuous_linear_map.tmp_seminormed_add_comm_group.to_pseudo_metric_space | def | continuous_linear_map.tmp_pseudo_metric_space | 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"
] | [
"pseudo_metric_space"
] | The `pseudo_metric_space` structure on `E →SL[σ₁₂] F` coming from
`continuous_linear_map.tmp_seminormed_add_comm_group`.
See Note [forgetful inheritance] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tmp_uniform_space : uniform_space (E →SL[σ₁₂] F) | continuous_linear_map.tmp_pseudo_metric_space.to_uniform_space | def | continuous_linear_map.tmp_uniform_space | 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"
] | [
"uniform_space"
] | The `uniform_space` structure on `E →SL[σ₁₂] F` coming from
`continuous_linear_map.tmp_seminormed_add_comm_group`.
See Note [forgetful inheritance] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tmp_topological_space : topological_space (E →SL[σ₁₂] F) | continuous_linear_map.tmp_uniform_space.to_topological_space | def | continuous_linear_map.tmp_topological_space | 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"
] | [
"topological_space"
] | The `topological_space` structure on `E →SL[σ₁₂] F` coming from
`continuous_linear_map.tmp_seminormed_add_comm_group`.
See Note [forgetful inheritance] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tmp_topological_add_group : topological_add_group (E →SL[σ₁₂] F) | infer_instance | lemma | continuous_linear_map.tmp_topological_add_group | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tmp_closed_ball_div_subset {a b : ℝ} (ha : 0 < a) (hb : 0 < b) :
closed_ball (0 : E →SL[σ₁₂] F) (a / b) ⊆
{f | ∀ x ∈ closed_ball (0 : E) b, f x ∈ closed_ball (0 : F) a} | begin
intros f hf x hx,
rw mem_closed_ball_zero_iff at ⊢ hf hx,
calc ‖f x‖
≤ ‖f‖ * ‖x‖ : le_op_norm _ _
... ≤ (a/b) * b : mul_le_mul hf hx (norm_nonneg _) (div_pos ha hb).le
... = a : div_mul_cancel a hb.ne.symm
end | lemma | continuous_linear_map.tmp_closed_ball_div_subset | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"div_mul_cancel",
"div_pos",
"mul_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tmp_topology_eq :
(continuous_linear_map.tmp_topological_space : topological_space (E →SL[σ₁₂] F)) =
continuous_linear_map.topological_space | begin
refine continuous_linear_map.tmp_topological_add_group.ext infer_instance
((@metric.nhds_basis_closed_ball _ continuous_linear_map.tmp_pseudo_metric_space 0).ext
(continuous_linear_map.has_basis_nhds_zero_of_basis metric.nhds_basis_closed_ball) _ _),
{ rcases normed_field.exists_norm_lt_one 𝕜 with ... | theorem | continuous_linear_map.tmp_topology_eq | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.has_basis_nhds_zero_of_basis",
"continuous_linear_map.tmp_closed_ball_div_subset",
"continuous_linear_map.tmp_pseudo_metric_space",
"continuous_linear_map.tmp_topological_space",
"div_mul_cancel",
"div_pos",
"le_mul_of_one_le_right",
"metric.nhds_basis_closed_ball",
"normed_fi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tmp_uniform_space_eq :
(continuous_linear_map.tmp_uniform_space : uniform_space (E →SL[σ₁₂] F)) =
continuous_linear_map.uniform_space | begin
rw [← @uniform_add_group.to_uniform_space_eq _ continuous_linear_map.tmp_uniform_space,
← @uniform_add_group.to_uniform_space_eq _ continuous_linear_map.uniform_space],
congr' 1,
exact continuous_linear_map.tmp_topology_eq
end | theorem | continuous_linear_map.tmp_uniform_space_eq | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.tmp_topology_eq",
"continuous_linear_map.tmp_uniform_space",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pseudo_metric_space : pseudo_metric_space (E →SL[σ₁₂] F) | continuous_linear_map.tmp_pseudo_metric_space.replace_uniformity
(congr_arg _ continuous_linear_map.tmp_uniform_space_eq.symm) | instance | continuous_linear_map.to_pseudo_metric_space | 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"
] | [
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_seminormed_add_comm_group : seminormed_add_comm_group (E →SL[σ₁₂] F) | { dist_eq := continuous_linear_map.tmp_seminormed_add_comm_group.dist_eq } | instance | continuous_linear_map.to_seminormed_add_comm_group | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"seminormed_add_comm_group"
] | Continuous linear maps themselves form a seminormed space with respect to
the operator norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_def (f : E →SL[σ₁₂] F) : ‖f‖₊ = Inf {c | ∀ x, ‖f x‖₊ ≤ c * ‖x‖₊} | begin
ext,
rw [nnreal.coe_Inf, coe_nnnorm, norm_def, nnreal.coe_image],
simp_rw [← nnreal.coe_le_coe, nnreal.coe_mul, coe_nnnorm, mem_set_of_eq, subtype.coe_mk,
exists_prop],
end | lemma | continuous_linear_map.nnnorm_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"
] | [
"exists_prop",
"nnreal.coe_Inf",
"nnreal.coe_image",
"nnreal.coe_le_coe",
"nnreal.coe_mul",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_nnnorm_le_bound (f : E →SL[σ₁₂] F) (M : ℝ≥0) (hM : ∀ x, ‖f x‖₊ ≤ M * ‖x‖₊) :
‖f‖₊ ≤ M | op_norm_le_bound f (zero_le M) hM | lemma | continuous_linear_map.op_nnnorm_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"
] | [] | If one controls the norm of every `A x`, then one controls the norm of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_nnnorm_le_bound' (f : E →SL[σ₁₂] F) (M : ℝ≥0) (hM : ∀ x, ‖x‖₊ ≠ 0 → ‖f x‖₊ ≤ M * ‖x‖₊) :
‖f‖₊ ≤ M | op_norm_le_bound' f (zero_le M) $ λ x hx, hM x $ by rwa [← nnreal.coe_ne_zero] | lemma | continuous_linear_map.op_nnnorm_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"
] | [
"nnreal.coe_ne_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_nnnorm_le_of_unit_nnnorm [normed_space ℝ E] [normed_space ℝ F] {f : E →L[ℝ] F} {C : ℝ≥0}
(hf : ∀ x, ‖x‖₊ = 1 → ‖f x‖₊ ≤ C) : ‖f‖₊ ≤ C | op_norm_le_of_unit_norm C.coe_nonneg $ λ x hx, hf x $ by rwa [← nnreal.coe_eq_one] | lemma | continuous_linear_map.op_nnnorm_le_of_unit_nnnorm | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"nnreal.coe_eq_one",
"normed_space"
] | For a continuous real linear map `f`, if one controls the norm of every `f x`, `‖x‖₊ = 1`, then
one controls the norm of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_nnnorm_le_of_lipschitz {f : E →SL[σ₁₂] F} {K : ℝ≥0} (hf : lipschitz_with K f) :
‖f‖₊ ≤ K | op_norm_le_of_lipschitz hf | theorem | continuous_linear_map.op_nnnorm_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_nnnorm_eq_of_bounds {φ : E →SL[σ₁₂] F} (M : ℝ≥0)
(h_above : ∀ x, ‖φ x‖ ≤ M*‖x‖) (h_below : ∀ N, (∀ x, ‖φ x‖₊ ≤ N*‖x‖₊) → M ≤ N) :
‖φ‖₊ = M | subtype.ext $ op_norm_eq_of_bounds (zero_le M) h_above $ subtype.forall'.mpr h_below | lemma | continuous_linear_map.op_nnnorm_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"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_normed_space {𝕜' : Type*} [normed_field 𝕜'] [normed_space 𝕜' F]
[smul_comm_class 𝕜₂ 𝕜' F] : normed_space 𝕜' (E →SL[σ₁₂] F) | ⟨op_norm_smul_le⟩ | instance | continuous_linear_map.to_normed_space | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"normed_field",
"normed_space",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_comp_le (f : E →SL[σ₁₂] F) : ‖h.comp f‖ ≤ ‖h‖ * ‖f‖ | (cInf_le bounds_bdd_below
⟨mul_nonneg (op_norm_nonneg _) (op_norm_nonneg _), λ x,
by { rw mul_assoc, exact h.le_op_norm_of_le (f.le_op_norm x) } ⟩) | lemma | continuous_linear_map.op_norm_comp_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"cInf_le",
"mul_assoc"
] | The operator norm is submultiplicative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_nnnorm_comp_le [ring_hom_isometric σ₁₃] (f : E →SL[σ₁₂] F) : ‖h.comp f‖₊ ≤ ‖h‖₊ * ‖f‖₊ | op_norm_comp_le h f | lemma | continuous_linear_map.op_nnnorm_comp_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"ring_hom_isometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_semi_normed_ring : semi_normed_ring (E →L[𝕜] E) | { norm_mul := λ f g, op_norm_comp_le f g,
.. continuous_linear_map.to_seminormed_add_comm_group, .. continuous_linear_map.ring } | instance | continuous_linear_map.to_semi_normed_ring | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.to_seminormed_add_comm_group",
"norm_mul",
"semi_normed_ring"
] | Continuous linear maps form a seminormed ring with respect to the operator norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_normed_algebra : normed_algebra 𝕜 (E →L[𝕜] E) | { .. continuous_linear_map.to_normed_space,
.. continuous_linear_map.algebra } | instance | continuous_linear_map.to_normed_algebra | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.to_normed_space",
"normed_algebra"
] | For a normed space `E`, continuous linear endomorphisms form a normed algebra with
respect to the operator norm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_op_nnnorm : ‖f x‖₊ ≤ ‖f‖₊ * ‖x‖₊ | f.le_op_norm x | theorem | continuous_linear_map.le_op_nnnorm | 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 | |
nndist_le_op_nnnorm (x y : E) : nndist (f x) (f y) ≤ ‖f‖₊ * nndist x y | dist_le_op_norm f x y | theorem | continuous_linear_map.nndist_le_op_nnnorm | 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 | |
lipschitz : lipschitz_with ‖f‖₊ f | add_monoid_hom_class.lipschitz_of_bound_nnnorm f _ f.le_op_nnnorm | theorem | continuous_linear_map.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"
] | continuous linear maps are Lipschitz continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_apply (x : E) : lipschitz_with ‖x‖₊ (λ f : E →SL[σ₁₂] F, f x) | lipschitz_with_iff_norm_sub_le.2 $ λ f g, ((f - g).le_op_norm x).trans_eq (mul_comm _ _) | theorem | continuous_linear_map.lipschitz_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"
] | [
"lipschitz_with",
"mul_comm"
] | Evaluation of a continuous linear map `f` at a point is Lipschitz continuous in `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_mul_lt_apply_of_lt_op_nnnorm (f : E →SL[σ₁₂] F) {r : ℝ≥0} (hr : r < ‖f‖₊) :
∃ x, r * ‖x‖₊ < ‖f x‖₊ | by simpa only [not_forall, not_le, set.mem_set_of] using not_mem_of_lt_cInf
(nnnorm_def f ▸ hr : r < Inf {c : ℝ≥0 | ∀ x, ‖f x‖₊ ≤ c * ‖x‖₊}) (order_bot.bdd_below _) | lemma | continuous_linear_map.exists_mul_lt_apply_of_lt_op_nnnorm | 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"
] | [
"not_forall",
"not_mem_of_lt_cInf",
"order_bot.bdd_below",
"set.mem_set_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mul_lt_of_lt_op_norm (f : E →SL[σ₁₂] F) {r : ℝ} (hr₀ : 0 ≤ r) (hr : r < ‖f‖) :
∃ x, r * ‖x‖ < ‖f x‖ | by { lift r to ℝ≥0 using hr₀, exact f.exists_mul_lt_apply_of_lt_op_nnnorm hr } | lemma | continuous_linear_map.exists_mul_lt_of_lt_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"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_lt_apply_of_lt_op_nnnorm {𝕜 𝕜₂ E F : Type*} [normed_add_comm_group E]
[seminormed_add_comm_group F] [densely_normed_field 𝕜] [nontrivially_normed_field 𝕜₂]
{σ₁₂ : 𝕜 →+* 𝕜₂} [normed_space 𝕜 E] [normed_space 𝕜₂ F] [ring_hom_isometric σ₁₂]
(f : E →SL[σ₁₂] F) {r : ℝ≥0} (hr : r < ‖f‖₊) : ∃ x : E, ‖x‖₊ <... | begin
obtain ⟨y, hy⟩ := f.exists_mul_lt_apply_of_lt_op_nnnorm hr,
have hy' : ‖y‖₊ ≠ 0 := nnnorm_ne_zero_iff.2
(λ heq, by simpa only [heq, nnnorm_zero, map_zero, not_lt_zero'] using hy),
have hfy : ‖f y‖₊ ≠ 0 := (zero_le'.trans_lt hy).ne',
rw [←inv_inv (‖f y‖₊), nnreal.lt_inv_iff_mul_lt (inv_ne_zero hfy), mu... | lemma | continuous_linear_map.exists_lt_apply_of_lt_op_nnnorm | 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"
] | [
"densely_normed_field",
"div_eq_mul_inv",
"inv_ne_zero",
"mul_assoc",
"mul_comm",
"nnnorm_smul",
"nnreal.lt_inv_iff_mul_lt",
"nontrivially_normed_field",
"normed_add_comm_group",
"normed_field.exists_lt_nnnorm_lt",
"normed_space",
"not_lt_zero'",
"ring_hom_isometric",
"seminormed_add_comm_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_lt_apply_of_lt_op_norm {𝕜 𝕜₂ E F : Type*} [normed_add_comm_group E]
[seminormed_add_comm_group F] [densely_normed_field 𝕜] [nontrivially_normed_field 𝕜₂]
{σ₁₂ : 𝕜 →+* 𝕜₂} [normed_space 𝕜 E] [normed_space 𝕜₂ F] [ring_hom_isometric σ₁₂]
(f : E →SL[σ₁₂] F) {r : ℝ} (hr : r < ‖f‖) : ∃ x : E, ‖x‖ < 1 ∧ r... | begin
by_cases hr₀ : r < 0,
{ exact ⟨0, by simpa using hr₀⟩, },
{ lift r to ℝ≥0 using not_lt.1 hr₀,
exact f.exists_lt_apply_of_lt_op_nnnorm hr, }
end | lemma | continuous_linear_map.exists_lt_apply_of_lt_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"
] | [
"densely_normed_field",
"lift",
"nontrivially_normed_field",
"normed_add_comm_group",
"normed_space",
"ring_hom_isometric",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_unit_ball_eq_nnnorm {𝕜 𝕜₂ E F : Type*} [normed_add_comm_group E]
[seminormed_add_comm_group F] [densely_normed_field 𝕜] [nontrivially_normed_field 𝕜₂]
{σ₁₂ : 𝕜 →+* 𝕜₂} [normed_space 𝕜 E] [normed_space 𝕜₂ F] [ring_hom_isometric σ₁₂]
(f : E →SL[σ₁₂] F) : Sup ((λ x, ‖f x‖₊) '' ball 0 1) = ‖f‖₊ | begin
refine cSup_eq_of_forall_le_of_forall_lt_exists_gt ((nonempty_ball.mpr zero_lt_one).image _)
_ (λ ub hub, _),
{ rintro - ⟨x, hx, rfl⟩,
simpa only [mul_one] using f.le_op_norm_of_le (mem_ball_zero_iff.1 hx).le },
{ obtain ⟨x, hx, hxf⟩ := f.exists_lt_apply_of_lt_op_nnnorm hub,
exact ⟨_, ⟨x, mem_ba... | lemma | continuous_linear_map.Sup_unit_ball_eq_nnnorm | 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"
] | [
"cSup_eq_of_forall_le_of_forall_lt_exists_gt",
"densely_normed_field",
"mul_one",
"nontrivially_normed_field",
"normed_add_comm_group",
"normed_space",
"ring_hom_isometric",
"seminormed_add_comm_group",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_unit_ball_eq_norm {𝕜 𝕜₂ E F : Type*} [normed_add_comm_group E]
[seminormed_add_comm_group F] [densely_normed_field 𝕜] [nontrivially_normed_field 𝕜₂]
{σ₁₂ : 𝕜 →+* 𝕜₂} [normed_space 𝕜 E] [normed_space 𝕜₂ F] [ring_hom_isometric σ₁₂]
(f : E →SL[σ₁₂] F) : Sup ((λ x, ‖f x‖) '' ball 0 1) = ‖f‖ | by simpa only [nnreal.coe_Sup, set.image_image] using nnreal.coe_eq.2 f.Sup_unit_ball_eq_nnnorm | lemma | continuous_linear_map.Sup_unit_ball_eq_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"
] | [
"densely_normed_field",
"nnreal.coe_Sup",
"nontrivially_normed_field",
"normed_add_comm_group",
"normed_space",
"ring_hom_isometric",
"seminormed_add_comm_group",
"set.image_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_closed_unit_ball_eq_nnnorm {𝕜 𝕜₂ E F : Type*} [normed_add_comm_group E]
[seminormed_add_comm_group F] [densely_normed_field 𝕜] [nontrivially_normed_field 𝕜₂]
{σ₁₂ : 𝕜 →+* 𝕜₂} [normed_space 𝕜 E] [normed_space 𝕜₂ F] [ring_hom_isometric σ₁₂]
(f : E →SL[σ₁₂] F) : Sup ((λ x, ‖f x‖₊) '' closed_ball 0 1) = ‖... | begin
have hbdd : ∀ y ∈ (λ x, ‖f x‖₊) '' closed_ball 0 1, y ≤ ‖f‖₊,
{ rintro - ⟨x, hx, rfl⟩, exact f.unit_le_op_norm x (mem_closed_ball_zero_iff.1 hx) },
refine le_antisymm (cSup_le ((nonempty_closed_ball.mpr zero_le_one).image _) hbdd) _,
rw ←Sup_unit_ball_eq_nnnorm,
exact cSup_le_cSup ⟨‖f‖₊, hbdd⟩ ((nonempt... | lemma | continuous_linear_map.Sup_closed_unit_ball_eq_nnnorm | 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"
] | [
"cSup_le",
"cSup_le_cSup",
"densely_normed_field",
"nontrivially_normed_field",
"normed_add_comm_group",
"normed_space",
"ring_hom_isometric",
"seminormed_add_comm_group",
"set.image_subset",
"zero_le_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_closed_unit_ball_eq_norm {𝕜 𝕜₂ E F : Type*} [normed_add_comm_group E]
[seminormed_add_comm_group F] [densely_normed_field 𝕜] [nontrivially_normed_field 𝕜₂]
{σ₁₂ : 𝕜 →+* 𝕜₂} [normed_space 𝕜 E] [normed_space 𝕜₂ F] [ring_hom_isometric σ₁₂]
(f : E →SL[σ₁₂] F) : Sup ((λ x, ‖f x‖) '' closed_ball 0 1) = ‖f‖ | by simpa only [nnreal.coe_Sup, set.image_image] using nnreal.coe_eq.2
f.Sup_closed_unit_ball_eq_nnnorm | lemma | continuous_linear_map.Sup_closed_unit_ball_eq_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"
] | [
"densely_normed_field",
"nnreal.coe_Sup",
"nontrivially_normed_field",
"normed_add_comm_group",
"normed_space",
"ring_hom_isometric",
"seminormed_add_comm_group",
"set.image_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_ext [ring_hom_isometric σ₁₃] (f : E →SL[σ₁₂] F) (g : E →SL[σ₁₃] G)
(h : ∀ x, ‖f x‖ = ‖g x‖) : ‖f‖ = ‖g‖ | op_norm_eq_of_bounds (norm_nonneg _) (λ x, by { rw h x, exact le_op_norm _ _ })
(λ c hc h₂, op_norm_le_bound _ hc (λ z, by { rw ←h z, exact h₂ z })) | lemma | continuous_linear_map.op_norm_ext | 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 | |
op_norm_le_bound₂ (f : E →SL[σ₁₃] F →SL[σ₂₃] G) {C : ℝ} (h0 : 0 ≤ C)
(hC : ∀ x y, ‖f x y‖ ≤ C * ‖x‖ * ‖y‖) :
‖f‖ ≤ C | f.op_norm_le_bound h0 $ λ x,
(f x).op_norm_le_bound (mul_nonneg h0 (norm_nonneg _)) $ hC x | theorem | continuous_linear_map.op_norm_le_bound₂ | 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 | |
le_op_norm₂ [ring_hom_isometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F) :
‖f x y‖ ≤ ‖f‖ * ‖x‖ * ‖y‖ | (f x).le_of_op_norm_le (f.le_op_norm x) y | theorem | continuous_linear_map.le_op_norm₂ | 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 | |
op_norm_prod (f : E →L[𝕜] Fₗ) (g : E →L[𝕜] Gₗ) : ‖f.prod g‖ = ‖(f, g)‖ | le_antisymm
(op_norm_le_bound _ (norm_nonneg _) $ λ x,
by simpa only [prod_apply, prod.norm_def, max_mul_of_nonneg, norm_nonneg]
using max_le_max (le_op_norm f x) (le_op_norm g x)) $
max_le
(op_norm_le_bound _ (norm_nonneg _) $ λ x, (le_max_left _ _).trans ((f.prod g).le_op_norm x))
(op_norm_le_bo... | lemma | continuous_linear_map.op_norm_prod | 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"
] | [
"max_le_max",
"max_mul_of_nonneg",
"prod.norm_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_nnnorm_prod (f : E →L[𝕜] Fₗ) (g : E →L[𝕜] Gₗ) : ‖f.prod g‖₊ = ‖(f, g)‖₊ | subtype.ext $ op_norm_prod f g | lemma | continuous_linear_map.op_nnnorm_prod | 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"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prodₗᵢ (R : Type*) [semiring R] [module R Fₗ] [module R Gₗ]
[has_continuous_const_smul R Fₗ] [has_continuous_const_smul R Gₗ]
[smul_comm_class 𝕜 R Fₗ] [smul_comm_class 𝕜 R Gₗ] :
(E →L[𝕜] Fₗ) × (E →L[𝕜] Gₗ) ≃ₗᵢ[R] (E →L[𝕜] Fₗ × Gₗ) | ⟨prodₗ R, λ ⟨f, g⟩, op_norm_prod f g⟩ | def | continuous_linear_map.prodₗᵢ | 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_continuous_const_smul",
"module",
"semiring",
"smul_comm_class"
] | `continuous_linear_map.prod` as a `linear_isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_norm_subsingleton [subsingleton E] : ‖f‖ = 0 | begin
refine le_antisymm _ (norm_nonneg _),
apply op_norm_le_bound _ rfl.ge,
intros x,
simp [subsingleton.elim x 0]
end | lemma | continuous_linear_map.op_norm_subsingleton | 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 | |
is_O_with_id (l : filter E) : is_O_with ‖f‖ l f (λ x, x) | is_O_with_of_le' _ f.le_op_norm | theorem | continuous_linear_map.is_O_with_id | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_id (l : filter E) : f =O[l] (λ x, x) | (f.is_O_with_id l).is_O | theorem | continuous_linear_map.is_O_id | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_comp [ring_hom_isometric σ₂₃] {α : Type*} (g : F →SL[σ₂₃] G) (f : α → F)
(l : filter α) :
is_O_with ‖g‖ l (λ x', g (f x')) f | (g.is_O_with_id ⊤).comp_tendsto le_top | theorem | continuous_linear_map.is_O_with_comp | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"filter",
"le_top",
"ring_hom_isometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_comp [ring_hom_isometric σ₂₃] {α : Type*} (g : F →SL[σ₂₃] G) (f : α → F)
(l : filter α) :
(λ x', g (f x')) =O[l] f | (g.is_O_with_comp f l).is_O | theorem | continuous_linear_map.is_O_comp | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"filter",
"ring_hom_isometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_sub (f : E →SL[σ₁₂] F) (l : filter E) (x : E) :
is_O_with ‖f‖ l (λ x', f (x' - x)) (λ x', x' - x) | f.is_O_with_comp _ l | theorem | continuous_linear_map.is_O_with_sub | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_sub (f : E →SL[σ₁₂] F) (l : filter E) (x : E) :
(λ x', f (x' - x)) =O[l] (λ x', x' - x) | f.is_O_comp _ l | theorem | continuous_linear_map.is_O_sub | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_to_continuous_linear_map_le (f : E →ₛₗᵢ[σ₁₂] F) :
‖f.to_continuous_linear_map‖ ≤ 1 | f.to_continuous_linear_map.op_norm_le_bound zero_le_one $ λ x, by simp | lemma | linear_isometry.norm_to_continuous_linear_map_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_continuous_norm_le (f : E →ₛₗ[σ₁₂] F) {C : ℝ} (hC : 0 ≤ C) (h : ∀x, ‖f x‖ ≤ C * ‖x‖) :
‖f.mk_continuous C h‖ ≤ C | continuous_linear_map.op_norm_le_bound _ hC h | lemma | linear_map.mk_continuous_norm_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.op_norm_le_bound"
] | If a continuous linear map is constructed from a linear 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 |
mk_continuous_norm_le' (f : E →ₛₗ[σ₁₂] F) {C : ℝ} (h : ∀x, ‖f x‖ ≤ C * ‖x‖) :
‖f.mk_continuous C h‖ ≤ max C 0 | continuous_linear_map.op_norm_le_bound _ (le_max_right _ _) $ λ x, (h x).trans $
mul_le_mul_of_nonneg_right (le_max_left _ _) (norm_nonneg x) | lemma | linear_map.mk_continuous_norm_le' | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.op_norm_le_bound",
"mul_le_mul_of_nonneg_right"
] | If a continuous linear map is constructed from a linear map via the constructor `mk_continuous`,
then its norm is bounded by the bound or zero if bound is negative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_continuous₂ (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) (C : ℝ)
(hC : ∀ x y, ‖f x y‖ ≤ C * ‖x‖ * ‖y‖) :
E →SL[σ₁₃] F →SL[σ₂₃] G | linear_map.mk_continuous
{ to_fun := λ x, (f x).mk_continuous (C * ‖x‖) (hC x),
map_add' := λ x y,
begin
ext z,
rw [continuous_linear_map.add_apply, mk_continuous_apply, mk_continuous_apply,
mk_continuous_apply, map_add, add_apply]
end,
map_smul' := λ c x,
begin
ext z,
... | def | linear_map.mk_continuous₂ | 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.add_apply",
"continuous_linear_map.smul_apply",
"linear_map.mk_continuous",
"max_mul_of_nonneg",
"zero_mul"
] | Create a bilinear map (represented as a map `E →L[𝕜] F →L[𝕜] G`) from the corresponding linear
map and a bound on the norm of the image. The linear map can be constructed using
`linear_map.mk₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_continuous₂_apply (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : ℝ}
(hC : ∀ x y, ‖f x y‖ ≤ C * ‖x‖ * ‖y‖) (x : E) (y : F) :
f.mk_continuous₂ C hC x y = f x y | rfl | lemma | linear_map.mk_continuous₂_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_continuous₂_norm_le' (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : ℝ}
(hC : ∀ x y, ‖f x y‖ ≤ C * ‖x‖ * ‖y‖) :
‖f.mk_continuous₂ C hC‖ ≤ max C 0 | mk_continuous_norm_le _ (le_max_iff.2 $ or.inr le_rfl) _ | lemma | linear_map.mk_continuous₂_norm_le' | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_continuous₂_norm_le (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : ℝ} (h0 : 0 ≤ C)
(hC : ∀ x y, ‖f x y‖ ≤ C * ‖x‖ * ‖y‖) :
‖f.mk_continuous₂ C hC‖ ≤ C | (f.mk_continuous₂_norm_le' hC).trans_eq $ max_eq_left h0 | lemma | linear_map.mk_continuous₂_norm_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : F →SL[σ₂₃] E →SL[σ₁₃] G | linear_map.mk_continuous₂
(linear_map.mk₂'ₛₗ σ₂₃ σ₁₃ (λ y x, f x y)
(λ x y z, (f z).map_add x y)
(λ c y x, (f x).map_smulₛₗ c y)
(λ z x y, by rw [f.map_add, add_apply])
(λ c y x, by rw [f.map_smulₛₗ, smul_apply]))
‖f‖
(λ y x, (f.le_op_norm₂ x y).trans_eq $ by rw mul_right_comm) | def | continuous_linear_map.flip | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"linear_map.mk_continuous₂",
"linear_map.mk₂'ₛₗ",
"mul_right_comm"
] | Flip the order of arguments of a continuous bilinear map.
For a version bundled as `linear_isometry_equiv`, see
`continuous_linear_map.flipL`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_norm_flip (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : ‖f‖ ≤ ‖flip f‖ | f.op_norm_le_bound₂ (norm_nonneg _) $ λ x y,
by { rw mul_right_comm, exact (flip f).le_op_norm₂ y x } | lemma | continuous_linear_map.le_norm_flip | 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_right_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip_apply (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F) : f.flip y x = f x y | rfl | lemma | continuous_linear_map.flip_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip_flip (f : E →SL[σ₁₃] F →SL[σ₂₃] G) :
f.flip.flip = f | by { ext, refl } | lemma | continuous_linear_map.flip_flip | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_norm_flip (f : E →SL[σ₁₃] F →SL[σ₂₃] G) :
‖f.flip‖ = ‖f‖ | le_antisymm (by simpa only [flip_flip] using le_norm_flip f.flip) (le_norm_flip f) | lemma | continuous_linear_map.op_norm_flip | 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 | |
flip_add (f g : E →SL[σ₁₃] F →SL[σ₂₃] G) :
(f + g).flip = f.flip + g.flip | rfl | lemma | continuous_linear_map.flip_add | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip_smul (c : 𝕜₃) (f : E →SL[σ₁₃] F →SL[σ₂₃] G) :
(c • f).flip = c • f.flip | rfl | lemma | continuous_linear_map.flip_smul | 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 | |
flipₗᵢ' : (E →SL[σ₁₃] F →SL[σ₂₃] G) ≃ₗᵢ[𝕜₃] (F →SL[σ₂₃] E →SL[σ₁₃] G) | { to_fun := flip,
inv_fun := flip,
map_add' := flip_add,
map_smul' := flip_smul,
left_inv := flip_flip,
right_inv := flip_flip,
norm_map' := op_norm_flip } | def | continuous_linear_map.flipₗᵢ' | 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"
] | [
"inv_fun"
] | Flip the order of arguments of a continuous bilinear map.
This is a version bundled as a `linear_isometry_equiv`.
For an unbundled version see `continuous_linear_map.flip`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
flipₗᵢ'_symm : (flipₗᵢ' E F G σ₂₃ σ₁₃).symm = flipₗᵢ' F E G σ₁₃ σ₂₃ | rfl | lemma | continuous_linear_map.flipₗᵢ'_symm | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_flipₗᵢ' : ⇑(flipₗᵢ' E F G σ₂₃ σ₁₃) = flip | rfl | lemma | continuous_linear_map.coe_flipₗᵢ' | 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 | |
flipₗᵢ : (E →L[𝕜] Fₗ →L[𝕜] Gₗ) ≃ₗᵢ[𝕜] (Fₗ →L[𝕜] E →L[𝕜] Gₗ) | { to_fun := flip,
inv_fun := flip,
map_add' := flip_add,
map_smul' := flip_smul,
left_inv := flip_flip,
right_inv := flip_flip,
norm_map' := op_norm_flip } | def | continuous_linear_map.flipₗᵢ | 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"
] | [
"inv_fun"
] | Flip the order of arguments of a continuous bilinear map.
This is a version bundled as a `linear_isometry_equiv`.
For an unbundled version see `continuous_linear_map.flip`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
flipₗᵢ_symm : (flipₗᵢ 𝕜 E Fₗ Gₗ).symm = flipₗᵢ 𝕜 Fₗ E Gₗ | rfl | lemma | continuous_linear_map.flipₗᵢ_symm | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_flipₗᵢ : ⇑(flipₗᵢ 𝕜 E Fₗ Gₗ) = flip | rfl | lemma | continuous_linear_map.coe_flipₗᵢ | 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 | |
apply' : E →SL[σ₁₂] (E →SL[σ₁₂] F) →L[𝕜₂] F | flip (id 𝕜₂ (E →SL[σ₁₂] F)) | def | continuous_linear_map.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"
] | [] | The continuous semilinear map obtained by applying a continuous semilinear map at a given
vector.
This is the continuous version of `linear_map.applyₗ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_apply' (v : E) (f : E →SL[σ₁₂] F) : apply' F σ₁₂ v f = f v | rfl | lemma | continuous_linear_map.apply_apply' | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply : E →L[𝕜] (E →L[𝕜] Fₗ) →L[𝕜] Fₗ | flip (id 𝕜 (E →L[𝕜] Fₗ)) | def | continuous_linear_map.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"
] | [] | The continuous semilinear map obtained by applying a continuous semilinear map at a given
vector.
This is the continuous version of `linear_map.applyₗ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_apply (v : E) (f : E →L[𝕜] Fₗ) : apply 𝕜 Fₗ v f = f v | rfl | lemma | continuous_linear_map.apply_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compSL : (F →SL[σ₂₃] G) →L[𝕜₃] (E →SL[σ₁₂] F) →SL[σ₂₃] (E →SL[σ₁₃] G) | linear_map.mk_continuous₂
(linear_map.mk₂'ₛₗ (ring_hom.id 𝕜₃) σ₂₃ comp add_comp smul_comp comp_add
(λ c f g, by { ext, simp only [continuous_linear_map.map_smulₛₗ, coe_smul', coe_comp',
function.comp_app, pi.smul_apply] }))
1 $ λ f g, by simpa only [one_mul] using op_norm_com... | def | continuous_linear_map.compSL | 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.map_smulₛₗ",
"linear_map.mk_continuous₂",
"linear_map.mk₂'ₛₗ",
"one_mul",
"pi.smul_apply",
"ring_hom.id"
] | Composition of continuous semilinear maps as a continuous semibilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_compSL_le : ‖compSL E F G σ₁₂ σ₂₃‖ ≤ 1 | linear_map.mk_continuous₂_norm_le _ zero_le_one _ | lemma | continuous_linear_map.norm_compSL_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"linear_map.mk_continuous₂_norm_le",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compSL_apply (f : F →SL[σ₂₃] G) (g : E →SL[σ₁₂] F) :
compSL E F G σ₁₂ σ₂₃ f g = f.comp g | rfl | lemma | continuous_linear_map.compSL_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous.const_clm_comp {X} [topological_space X] {f : X → E →SL[σ₁₂] F}
(hf : continuous f) (g : F →SL[σ₂₃] G) : continuous (λ x, g.comp (f x) : X → E →SL[σ₁₃] G) | (compSL E F G σ₁₂ σ₂₃ g).continuous.comp hf | lemma | continuous.const_clm_comp | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous",
"continuous.comp",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous.clm_comp_const {X} [topological_space X] {g : X → F →SL[σ₂₃] G}
(hg : continuous g) (f : E →SL[σ₁₂] F) : continuous (λ x, (g x).comp f : X → E →SL[σ₁₃] G) | (@continuous_linear_map.flip _ _ _ _ _ (E →SL[σ₁₃] G) _ _ _ _ _ _ _ _ _ _ _ _ _
(compSL E F G σ₁₂ σ₂₃) f).continuous.comp hg | lemma | continuous.clm_comp_const | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous",
"continuous.comp",
"continuous_linear_map.flip",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compL : (Fₗ →L[𝕜] Gₗ) →L[𝕜] (E →L[𝕜] Fₗ) →L[𝕜] (E →L[𝕜] Gₗ) | compSL E Fₗ Gₗ (ring_hom.id 𝕜) (ring_hom.id 𝕜) | def | continuous_linear_map.compL | 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.id"
] | Composition of continuous linear maps as a continuous bilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_compL_le : ‖compL 𝕜 E Fₗ Gₗ‖ ≤ 1 | norm_compSL_le _ _ _ _ _ | lemma | continuous_linear_map.norm_compL_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compL_apply (f : Fₗ →L[𝕜] Gₗ) (g : E →L[𝕜] Fₗ) : compL 𝕜 E Fₗ Gₗ f g = f.comp g | rfl | lemma | continuous_linear_map.compL_apply | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
precompR (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : E →L[𝕜] (Eₗ →L[𝕜] Fₗ) →L[𝕜] (Eₗ →L[𝕜] Gₗ) | (compL 𝕜 Eₗ Fₗ Gₗ).comp L | def | continuous_linear_map.precompR | 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"
] | [] | Apply `L(x,-)` pointwise to bilinear maps, as a continuous bilinear map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precompL (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : (Eₗ →L[𝕜] E) →L[𝕜] Fₗ →L[𝕜] (Eₗ →L[𝕜] Gₗ) | (precompR Eₗ (flip L)).flip | def | continuous_linear_map.precompL | 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"
] | [] | Apply `L(-,y)` pointwise to bilinear maps, as a continuous bilinear map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_precompR_le (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : ‖precompR Eₗ L‖ ≤ ‖L‖ | calc
‖precompR Eₗ L‖ ≤ ‖compL 𝕜 Eₗ Fₗ Gₗ‖ * ‖L‖ : op_norm_comp_le _ _
... ≤ 1 * ‖L‖ : mul_le_mul_of_nonneg_right (norm_compL_le _ _ _ _) (norm_nonneg _)
... = ‖L‖ : by rw one_mul | lemma | continuous_linear_map.norm_precompR_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"mul_le_mul_of_nonneg_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_precompL_le (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : ‖precompL Eₗ L‖ ≤ ‖L‖ | by { rw [precompL, op_norm_flip, ← op_norm_flip L], exact norm_precompR_le _ L.flip } | lemma | continuous_linear_map.norm_precompL_le | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mapL : ((M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄)) →L[𝕜] ((M₁ × M₃) →L[𝕜] (M₂ × M₄)) | continuous_linear_map.copy
(have Φ₁ : (M₁ →L[𝕜] M₂) →L[𝕜] (M₁ →L[𝕜] M₂ × M₄), from
continuous_linear_map.compL 𝕜 M₁ M₂ (M₂ × M₄) (continuous_linear_map.inl 𝕜 M₂ M₄),
have Φ₂ : (M₃ →L[𝕜] M₄) →L[𝕜] (M₃ →L[𝕜] M₂ × M₄), from
continuous_linear_map.compL 𝕜 M₃ M₄ (M₂ × M₄) (continuous_linear_map.inr 𝕜 M₂ M₄),
ha... | def | continuous_linear_map.prod_mapL | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous_linear_map.compL",
"continuous_linear_map.copy",
"continuous_linear_map.ext",
"continuous_linear_map.fst",
"continuous_linear_map.inl",
"continuous_linear_map.inr",
"continuous_linear_map.snd",
"prod.mk.inj_iff",
"prod_map"
] | `continuous_linear_map.prod_map` as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_mapL_apply (p : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄)) :
continuous_linear_map.prod_mapL 𝕜 M₁ M₂ M₃ M₄ p = p.1.prod_map p.2 | rfl | lemma | continuous_linear_map.prod_mapL_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"
] | [
"continuous_linear_map.prod_mapL"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous.prod_mapL {f : X → M₁ →L[𝕜] M₂} {g : X → M₃ →L[𝕜] M₄}
(hf : continuous f) (hg : continuous g) : continuous (λ x, (f x).prod_map (g x)) | (prod_mapL 𝕜 M₁ M₂ M₃ M₄).continuous.comp (hf.prod_mk hg) | lemma | continuous.prod_mapL | analysis.normed_space | src/analysis/normed_space/operator_norm.lean | [
"algebra.algebra.tower",
"analysis.asymptotics.asymptotics",
"analysis.normed_space.continuous_linear_map",
"analysis.normed_space.linear_isometry",
"topology.algebra.module.strong_topology"
] | [
"continuous",
"continuous.comp",
"prod_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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