# Documentation

Mathlib.Analysis.NormedSpace.OperatorNorm

# Operator norm on the space of continuous linear maps #

Define the operator norm on the space of continuous (semi)linear maps between normed spaces, and prove its basic properties. In particular, show that this space is itself a normed space.

Since a lot of elementary properties don't require ‖x‖ = 0 → x = 0 we start setting up the theory for SeminormedAddCommGroup and we specialize to NormedAddCommGroup at the end.

Note that most of statements that apply to semilinear maps only hold when the ring homomorphism is isometric, as expressed by the typeclass [RingHomIsometric σ].

theorem norm_image_of_norm_zero {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} {𝓕 : Type u_10} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [SemilinearMapClass 𝓕 σ₁₂ E F] (f : 𝓕) (hf : ) {x : E} (hx : x = 0) :
f x = 0

If ‖x‖ = 0 and f is continuous then ‖f x‖ = 0.

theorem SemilinearMapClass.bound_of_shell_semi_normed {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} {𝓕 : Type u_10} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] [SemilinearMapClass 𝓕 σ₁₂ E F] (f : 𝓕) {ε : } {C : } (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < c) (hf : ∀ (x : E), ε / c xx < εf x C * x) {x : E} (hx : x 0) :
f x C * x
theorem SemilinearMapClass.bound_of_continuous {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} {𝓕 : Type u_10} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] [SemilinearMapClass 𝓕 σ₁₂ E F] (f : 𝓕) (hf : ) :
C, 0 < C ∀ (x : E), f x C * x

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 control for the original element follows by rescaling.

theorem ContinuousLinearMap.bound {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) :
C, 0 < C ∀ (x : E), f x C * x
def LinearIsometry.toSpanSingleton (𝕜 : Type u_1) (E : Type u_4) [] {v : E} (hv : v = 1) :
𝕜 →ₗᵢ[𝕜] E

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.

Instances For
@[simp]
theorem LinearIsometry.toSpanSingleton_apply {𝕜 : Type u_1} {E : Type u_4} [] {v : E} (hv : v = 1) (a : 𝕜) :
↑() a = a v
@[simp]
theorem LinearIsometry.coe_toSpanSingleton {𝕜 : Type u_1} {E : Type u_4} [] {v : E} (hv : v = 1) :
().toLinearMap =
def ContinuousLinearMap.opNorm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) :

The operator norm of a continuous linear map is the inf of all its bounds.

Instances For
instance ContinuousLinearMap.hasOpNorm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} :
Norm (E →SL[σ₁₂] F)
theorem ContinuousLinearMap.norm_def {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) :
f = sInf {c | 0 c ∀ (x : E), f x c * x}
theorem ContinuousLinearMap.bounds_nonempty {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] {f : E →SL[σ₁₂] F} :
c, c {c | 0 c ∀ (x : E), f x c * x}
theorem ContinuousLinearMap.bounds_bddBelow {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {f : E →SL[σ₁₂] F} :
BddBelow {c | 0 c ∀ (x : E), f x c * x}
theorem ContinuousLinearMap.isLeast_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) :
IsLeast {c | 0 c ∀ (x : E), f x c * x} f
theorem ContinuousLinearMap.op_norm_le_bound {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) {M : } (hMp : 0 M) (hM : ∀ (x : E), f x M * x) :

If one controls the norm of every A x, then one controls the norm of A.

theorem ContinuousLinearMap.op_norm_le_bound' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) {M : } (hMp : 0 M) (hM : ∀ (x : E), x 0f x M * x) :

If one controls the norm of every A x, ‖x‖ ≠ 0, then one controls the norm of A.

theorem ContinuousLinearMap.op_norm_le_of_lipschitz {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {f : E →SL[σ₁₂] F} {K : NNReal} (hf : ) :
f K
theorem ContinuousLinearMap.op_norm_eq_of_bounds {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {φ : E →SL[σ₁₂] F} {M : } (M_nonneg : 0 M) (h_above : ∀ (x : E), φ x M * x) (h_below : ∀ (N : ), N 0(∀ (x : E), φ x N * x) → M N) :
φ = M
theorem ContinuousLinearMap.op_norm_neg {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) :
theorem ContinuousLinearMap.op_norm_nonneg {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) :
theorem ContinuousLinearMap.op_norm_zero {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} :

The norm of the 0 operator is 0.

theorem ContinuousLinearMap.norm_id_le {𝕜 : Type u_1} {E : Type u_4} [] :

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.

theorem ContinuousLinearMap.le_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (x : E) :

The fundamental property of the operator norm: ‖f x‖ ≤ ‖f‖ * ‖x‖.

theorem ContinuousLinearMap.dist_le_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (x : E) (y : E) :
dist (f x) (f y) f * dist x y
theorem ContinuousLinearMap.le_of_op_norm_le_of_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) {x : E} {a : } {b : } (hf : f a) (hx : x b) :
f x a * b
theorem ContinuousLinearMap.le_op_norm_of_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) {c : } {x : E} (h : x c) :
f x f * c
theorem ContinuousLinearMap.le_of_op_norm_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) {c : } (h : f c) (x : E) :
f x c * x
theorem ContinuousLinearMap.op_norm_le_iff {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] {f : E →SL[σ₁₂] F} {M : } (hMp : 0 M) :
f M ∀ (x : E), f x M * x
theorem ContinuousLinearMap.ratio_le_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (x : E) :
theorem ContinuousLinearMap.unit_le_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (x : E) :
x 1f x f

The image of the unit ball under a continuous linear map is bounded.

theorem ContinuousLinearMap.op_norm_le_of_shell {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] {f : E →SL[σ₁₂] F} {ε : } {C : } (ε_pos : 0 < ε) (hC : 0 C) {c : 𝕜} (hc : 1 < c) (hf : ∀ (x : E), ε / c xx < εf x C * x) :
theorem ContinuousLinearMap.op_norm_le_of_ball {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] {f : E →SL[σ₁₂] F} {ε : } {C : } (ε_pos : 0 < ε) (hC : 0 C) (hf : ∀ (x : E), x f x C * x) :
theorem ContinuousLinearMap.op_norm_le_of_nhds_zero {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] {f : E →SL[σ₁₂] F} {C : } (hC : 0 C) (hf : ∀ᶠ (x : E) in nhds 0, f x C * x) :
theorem ContinuousLinearMap.op_norm_le_of_shell' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] {f : E →SL[σ₁₂] F} {ε : } {C : } (ε_pos : 0 < ε) (hC : 0 C) {c : 𝕜} (hc : c < 1) (hf : ∀ (x : E), ε * c xx < εf x C * x) :
theorem ContinuousLinearMap.op_norm_le_of_unit_norm {E : Type u_4} {F : Type u_6} [] [] {f : E →L[] F} {C : } (hC : 0 C) (hf : ∀ (x : E), x = 1f x C) :

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.

theorem ContinuousLinearMap.op_norm_add_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (g : E →SL[σ₁₂] F) :

The operator norm satisfies the triangle inequality.

theorem ContinuousLinearMap.norm_id_of_nontrivial_seminorm {𝕜 : Type u_1} {E : Type u_4} [] (h : x, x 0) :

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.)

theorem ContinuousLinearMap.op_norm_smul_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] {𝕜' : Type u_11} [] [NormedSpace 𝕜' F] [SMulCommClass 𝕜₂ 𝕜' F] (c : 𝕜') (f : E →SL[σ₁₂] F) :
def ContinuousLinearMap.tmpSeminormedAddCommGroup {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] :

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 ContinuousLinearMap.topologicalSpace to avoid diamond issues. See Note [forgetful inheritance]

Instances For
def ContinuousLinearMap.tmpPseudoMetricSpace {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] :

The PseudoMetricSpace structure on E →SL[σ₁₂] F coming from ContinuousLinearMap.tmpSeminormedAddCommGroup. See Note [forgetful inheritance]

Instances For
def ContinuousLinearMap.tmpUniformSpace {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] :
UniformSpace (E →SL[σ₁₂] F)

The UniformSpace structure on E →SL[σ₁₂] F coming from ContinuousLinearMap.tmpSeminormedAddCommGroup. See Note [forgetful inheritance]

Instances For
def ContinuousLinearMap.tmpTopologicalSpace {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] :

The TopologicalSpace structure on E →SL[σ₁₂] F coming from ContinuousLinearMap.tmpSeminormedAddCommGroup. See Note [forgetful inheritance]

Instances For
theorem ContinuousLinearMap.tmpTopologicalAddGroup {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] :
theorem ContinuousLinearMap.tmp_closedBall_div_subset {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] {a : } {b : } (ha : 0 < a) (hb : 0 < b) :
Metric.closedBall 0 (a / b) {f | ∀ (x : E), x f x }
theorem ContinuousLinearMap.tmp_topology_eq {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] :
ContinuousLinearMap.tmpTopologicalSpace = ContinuousLinearMap.topologicalSpace
theorem ContinuousLinearMap.tmpUniformSpace_eq {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] :
ContinuousLinearMap.tmpUniformSpace = ContinuousLinearMap.uniformSpace
instance ContinuousLinearMap.toPseudoMetricSpace {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] :
instance ContinuousLinearMap.toSeminormedAddCommGroup {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] :

Continuous linear maps themselves form a seminormed space with respect to the operator norm.

theorem ContinuousLinearMap.nnnorm_def {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) :
f‖₊ = sInf {c | ∀ (x : E), f x‖₊ c * x‖₊}
theorem ContinuousLinearMap.op_nnnorm_le_bound {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (M : NNReal) (hM : ∀ (x : E), f x‖₊ M * x‖₊) :

If one controls the norm of every A x, then one controls the norm of A.

theorem ContinuousLinearMap.op_nnnorm_le_bound' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (M : NNReal) (hM : ∀ (x : E), x‖₊ 0f x‖₊ M * x‖₊) :

If one controls the norm of every A x, ‖x‖₊ ≠ 0, then one controls the norm of A.

theorem ContinuousLinearMap.op_nnnorm_le_of_unit_nnnorm {E : Type u_4} {F : Type u_6} [] [] {f : E →L[] F} {C : NNReal} (hf : ∀ (x : E), x‖₊ = 1f x‖₊ C) :

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.

theorem ContinuousLinearMap.op_nnnorm_le_of_lipschitz {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] {f : E →SL[σ₁₂] F} {K : NNReal} (hf : ) :
theorem ContinuousLinearMap.op_nnnorm_eq_of_bounds {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] {φ : E →SL[σ₁₂] F} (M : NNReal) (h_above : ∀ (x : E), φ x M * x) (h_below : ∀ (N : NNReal), (∀ (x : E), φ x‖₊ N * x‖₊) → M N) :
theorem ContinuousLinearMap.op_nnnorm_le_iff {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] {f : E →SL[σ₁₂] F} {C : NNReal} :
f‖₊ C ∀ (x : E), f x‖₊ C * x‖₊
theorem ContinuousLinearMap.isLeast_op_nnnorm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) :
IsLeast {C | ∀ (x : E), f x‖₊ C * x‖₊} f‖₊
instance ContinuousLinearMap.toNormedSpace {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] {𝕜' : Type u_11} [] [NormedSpace 𝕜' F] [SMulCommClass 𝕜₂ 𝕜' F] :
NormedSpace 𝕜' (E →SL[σ₁₂] F)
theorem ContinuousLinearMap.op_norm_comp_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [] [] (h : F →SL[σ₂₃] G) (f : E →SL[σ₁₂] F) :

The operator norm is submultiplicative.

theorem ContinuousLinearMap.op_norm_comp_le' {𝕜 : Type u_1} {E : Type u_4} {Eₗ : Type u_5} {Fₗ : Type u_7} [] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] (h : Eₗ →L[𝕜] Fₗ) (f : E →L[𝕜] Eₗ) :

The operator norm is submultiplicative.

theorem ContinuousLinearMap.op_nnnorm_comp_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [] [] (h : F →SL[σ₂₃] G) [] (f : E →SL[σ₁₂] F) :
instance ContinuousLinearMap.toSemiNormedRing {𝕜 : Type u_1} {E : Type u_4} [] :

Continuous linear maps form a seminormed ring with respect to the operator norm.

instance ContinuousLinearMap.toNormedAlgebra {𝕜 : Type u_1} {E : Type u_4} [] :
NormedAlgebra 𝕜 (E →L[𝕜] E)

For a normed space E, continuous linear endomorphisms form a normed algebra with respect to the operator norm.

theorem ContinuousLinearMap.le_op_nnnorm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (x : E) :
theorem ContinuousLinearMap.nndist_le_op_nnnorm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (x : E) (y : E) :
nndist (f x) (f y) f‖₊ * nndist x y
theorem ContinuousLinearMap.lipschitz {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) :

continuous linear maps are Lipschitz continuous.

theorem ContinuousLinearMap.lipschitz_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (x : E) :
LipschitzWith x‖₊ fun f => f x

Evaluation of a continuous linear map f at a point is Lipschitz continuous in f.

theorem ContinuousLinearMap.exists_mul_lt_apply_of_lt_op_nnnorm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) {r : NNReal} (hr : r < f‖₊) :
x, r * x‖₊ < f x‖₊
theorem ContinuousLinearMap.exists_mul_lt_of_lt_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) {r : } (hr₀ : 0 r) (hr : r < f) :
x, r * x < f x
theorem ContinuousLinearMap.exists_lt_apply_of_lt_op_nnnorm {𝕜 : Type u_11} {𝕜₂ : Type u_12} {E : Type u_13} {F : Type u_14} {σ₁₂ : 𝕜 →+* 𝕜₂} [] [NormedSpace 𝕜₂ F] [] (f : E →SL[σ₁₂] F) {r : NNReal} (hr : r < f‖₊) :
x, x‖₊ < 1 r < f x‖₊
theorem ContinuousLinearMap.exists_lt_apply_of_lt_op_norm {𝕜 : Type u_11} {𝕜₂ : Type u_12} {E : Type u_13} {F : Type u_14} {σ₁₂ : 𝕜 →+* 𝕜₂} [] [NormedSpace 𝕜₂ F] [] (f : E →SL[σ₁₂] F) {r : } (hr : r < f) :
x, x < 1 r < f x
theorem ContinuousLinearMap.sSup_unit_ball_eq_nnnorm {𝕜 : Type u_11} {𝕜₂ : Type u_12} {E : Type u_13} {F : Type u_14} {σ₁₂ : 𝕜 →+* 𝕜₂} [] [NormedSpace 𝕜₂ F] [] (f : E →SL[σ₁₂] F) :
sSup ((fun x => f x‖₊) '' ) = f‖₊
theorem ContinuousLinearMap.sSup_unit_ball_eq_norm {𝕜 : Type u_11} {𝕜₂ : Type u_12} {E : Type u_13} {F : Type u_14} {σ₁₂ : 𝕜 →+* 𝕜₂} [] [NormedSpace 𝕜₂ F] [] (f : E →SL[σ₁₂] F) :
sSup ((fun x => f x) '' ) = f
theorem ContinuousLinearMap.sSup_closed_unit_ball_eq_nnnorm {𝕜 : Type u_11} {𝕜₂ : Type u_12} {E : Type u_13} {F : Type u_14} {σ₁₂ : 𝕜 →+* 𝕜₂} [] [NormedSpace 𝕜₂ F] [] (f : E →SL[σ₁₂] F) :
sSup ((fun x => f x‖₊) '' ) = f‖₊
theorem ContinuousLinearMap.sSup_closed_unit_ball_eq_norm {𝕜 : Type u_11} {𝕜₂ : Type u_12} {E : Type u_13} {F : Type u_14} {σ₁₂ : 𝕜 →+* 𝕜₂} [] [NormedSpace 𝕜₂ F] [] (f : E →SL[σ₁₂] F) :
sSup ((fun x => f x) '' ) = f
theorem ContinuousLinearMap.op_norm_ext {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₁₃ : 𝕜 →+* 𝕜₃} [] (f : E →SL[σ₁₂] F) (g : E →SL[σ₁₃] G) (h : ∀ (x : E), f x = g x) :
theorem ContinuousLinearMap.op_norm_le_bound₂ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) {C : } (h0 : 0 C) (hC : ∀ (x : E) (y : F), ↑(f x) y C * x * y) :
theorem ContinuousLinearMap.le_op_norm₂ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] [] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F) :
↑(f x) y f * x * y
theorem ContinuousLinearMap.le_of_op_norm₂_le_of_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] [] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) {x : E} {y : F} {a : } {b : } {c : } (hf : f a) (hx : x b) (hy : y c) :
↑(f x) y a * b * c
@[simp]
theorem ContinuousLinearMap.op_norm_prod {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : E →L[𝕜] Fₗ) (g : E →L[𝕜] Gₗ) :
= (f, g)
@[simp]
theorem ContinuousLinearMap.op_nnnorm_prod {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : E →L[𝕜] Fₗ) (g : E →L[𝕜] Gₗ) :
= (f, g)‖₊
def ContinuousLinearMap.prodₗᵢ {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (R : Type u_11) [] [Module R Fₗ] [Module R Gₗ] [] [] [SMulCommClass 𝕜 R Fₗ] [SMulCommClass 𝕜 R Gₗ] :
(E →L[𝕜] Fₗ) × (E →L[𝕜] Gₗ) ≃ₗᵢ[R] E →L[𝕜] Fₗ × Gₗ

ContinuousLinearMap.prod as a LinearIsometryEquiv.

Instances For
@[simp]
theorem ContinuousLinearMap.op_norm_subsingleton {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) [] :
theorem ContinuousLinearMap.isBigOWith_id {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (l : ) :
Asymptotics.IsBigOWith f l f fun x => x
theorem ContinuousLinearMap.isBigO_id {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (l : ) :
f =O[l] fun x => x
theorem ContinuousLinearMap.isBigOWith_comp {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {F : Type u_6} {G : Type u_8} [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} [] {α : Type u_11} (g : F →SL[σ₂₃] G) (f : αF) (l : ) :
Asymptotics.IsBigOWith g l (fun x' => g (f x')) f
theorem ContinuousLinearMap.isBigO_comp {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {F : Type u_6} {G : Type u_8} [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} [] {α : Type u_11} (g : F →SL[σ₂₃] G) (f : αF) (l : ) :
(fun x' => g (f x')) =O[l] f
theorem ContinuousLinearMap.isBigOWith_sub {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (l : ) (x : E) :
Asymptotics.IsBigOWith f l (fun x' => f (x' - x)) fun x' => x' - x
theorem ContinuousLinearMap.isBigO_sub {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →SL[σ₁₂] F) (l : ) (x : E) :
(fun x' => f (x' - x)) =O[l] fun x' => x' - x
theorem LinearIsometry.norm_toContinuousLinearMap_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →ₛₗᵢ[σ₁₂] F) :
theorem LinearMap.mkContinuous_norm_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ₁₂] F) {C : } (hC : 0 C) (h : ∀ (x : E), f x C * x) :

If a continuous linear map is constructed from a linear map via the constructor mkContinuous, then its norm is bounded by the bound given to the constructor if it is nonnegative.

theorem LinearMap.mkContinuous_norm_le' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ₁₂] F) {C : } (h : ∀ (x : E), f x C * x) :

If a continuous linear map is constructed from a linear map via the constructor mkContinuous, then its norm is bounded by the bound or zero if bound is negative.

theorem LinearMap.norm_mkContinuous₂_aux {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) (C : ) (h : ∀ (x : E) (y : F), ↑(f x) y C * x * y) (x : E) :
def LinearMap.mkContinuousOfExistsBound₂ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) (h : C, ∀ (x : E) (y : F), ↑(f x) y C * x * y) :
E →SL[σ₁₃] F →SL[σ₂₃] G

Create a bilinear map (represented as a map E →L[𝕜] F →L[𝕜] G) from the corresponding linear map and existence of a bound on the norm of the image. The linear map can be constructed using LinearMap.mk₂.

If you have an explicit bound, use LinearMap.mkContinuous₂ instead, as a norm estimate will follow automatically in LinearMap.mkContinuous₂_norm_le.

Instances For
def LinearMap.mkContinuous₂ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) (C : ) (hC : ∀ (x : E) (y : F), ↑(f x) y C * x * y) :
E →SL[σ₁₃] F →SL[σ₂₃] G

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 LinearMap.mk₂. Lemmas LinearMap.mkContinuous₂_norm_le' and LinearMap.mkContinuous₂_norm_le provide estimates on the norm of an operator constructed using this function.

Instances For
@[simp]
theorem LinearMap.mkContinuous₂_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : } (hC : ∀ (x : E) (y : F), ↑(f x) y C * x * y) (x : E) (y : F) :
↑(↑() x) y = ↑(f x) y
theorem LinearMap.mkContinuous₂_norm_le' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : } (hC : ∀ (x : E) (y : F), ↑(f x) y C * x * y) :
theorem LinearMap.mkContinuous₂_norm_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : } (h0 : 0 C) (hC : ∀ (x : E) (y : F), ↑(f x) y C * x * y) :
def ContinuousLinearMap.flip {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] [] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) :
F →SL[σ₂₃] E →SL[σ₁₃] G

Flip the order of arguments of a continuous bilinear map. For a version bundled as LinearIsometryEquiv, see ContinuousLinearMap.flipL.

Instances For
@[simp]
theorem ContinuousLinearMap.flip_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] [] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F) :
↑(↑() y) x = ↑(f x) y
@[simp]
theorem ContinuousLinearMap.flip_flip {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] [] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) :
@[simp]
theorem ContinuousLinearMap.op_norm_flip {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] [] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) :
@[simp]
theorem ContinuousLinearMap.flip_add {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] [] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (g : E →SL[σ₁₃] F →SL[σ₂₃] G) :
@[simp]
theorem ContinuousLinearMap.flip_smul {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] [] (c : 𝕜₃) (f : E →SL[σ₁₃] F →SL[σ₂₃] G) :
def ContinuousLinearMap.flipₗᵢ' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} (E : Type u_4) (F : Type u_6) (G : Type u_8) [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] (σ₂₃ : 𝕜₂ →+* 𝕜₃) (σ₁₃ : 𝕜 →+* 𝕜₃) [] [] :
(E →SL[σ₁₃] F →SL[σ₂₃] G) ≃ₗᵢ[𝕜₃] F →SL[σ₂₃] E →SL[σ₁₃] G

Flip the order of arguments of a continuous bilinear map. This is a version bundled as a LinearIsometryEquiv. For an unbundled version see ContinuousLinearMap.flip.

Instances For
@[simp]
theorem ContinuousLinearMap.flipₗᵢ'_symm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] [] :
@[simp]
theorem ContinuousLinearMap.coe_flipₗᵢ' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [] [] :
↑(ContinuousLinearMap.flipₗᵢ' E F G σ₂₃ σ₁₃) = ContinuousLinearMap.flip
def ContinuousLinearMap.flipₗᵢ (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] :
(E →L[𝕜] Fₗ →L[𝕜] Gₗ) ≃ₗᵢ[𝕜] Fₗ →L[𝕜] E →L[𝕜] Gₗ

Flip the order of arguments of a continuous bilinear map. This is a version bundled as a LinearIsometryEquiv. For an unbundled version see ContinuousLinearMap.flip.

Instances For
@[simp]
theorem ContinuousLinearMap.flipₗᵢ_symm {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] :
=
@[simp]
theorem ContinuousLinearMap.coe_flipₗᵢ {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] :
↑() = ContinuousLinearMap.flip
noncomputable def ContinuousLinearMap.apply' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} (F : Type u_6) [] [NormedSpace 𝕜₂ F] (σ₁₂ : 𝕜 →+* 𝕜₂) [] :
E →SL[σ₁₂] (E →SL[σ₁₂] F) →L[𝕜₂] F

The continuous semilinear map obtained by applying a continuous semilinear map at a given vector.

This is the continuous version of LinearMap.applyₗ.

Instances For
@[simp]
theorem ContinuousLinearMap.apply_apply' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (v : E) (f : E →SL[σ₁₂] F) :
↑(↑() v) f = f v
noncomputable def ContinuousLinearMap.apply (𝕜 : Type u_1) {E : Type u_4} (Fₗ : Type u_7) [] [NormedSpace 𝕜 Fₗ] :
E →L[𝕜] (E →L[𝕜] Fₗ) →L[𝕜] Fₗ

The continuous semilinear map obtained by applying a continuous semilinear map at a given vector.

This is the continuous version of LinearMap.applyₗ.

Instances For
@[simp]
theorem ContinuousLinearMap.apply_apply {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [] [NormedSpace 𝕜 Fₗ] (v : E) (f : E →L[𝕜] Fₗ) :
↑(↑() v) f = f v
noncomputable def ContinuousLinearMap.compSL {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} (E : Type u_4) (F : Type u_6) (G : Type u_8) [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] (σ₁₂ : 𝕜 →+* 𝕜₂) (σ₂₃ : 𝕜₂ →+* 𝕜₃) {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [] [] [] :
(F →SL[σ₂₃] G) →L[𝕜₃] (E →SL[σ₁₂] F) →SL[σ₂₃] E →SL[σ₁₃] G

Composition of continuous semilinear maps as a continuous semibilinear map.

Instances For
theorem ContinuousLinearMap.norm_compSL_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} (E : Type u_4) (F : Type u_6) (G : Type u_8) [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] (σ₁₂ : 𝕜 →+* 𝕜₂) (σ₂₃ : 𝕜₂ →+* 𝕜₃) {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [] [] [] :
ContinuousLinearMap.compSL E F G σ₁₂ σ₂₃ 1
@[simp]
theorem ContinuousLinearMap.compSL_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [] [] [] (f : F →SL[σ₂₃] G) (g : E →SL[σ₁₂] F) :
↑(↑(ContinuousLinearMap.compSL E F G σ₁₂ σ₂₃) f) g =
theorem Continuous.const_clm_comp {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [] [] [] {X : Type u_11} [] {f : XE →SL[σ₁₂] F} (hf : ) (g : F →SL[σ₂₃] G) :
theorem Continuous.clm_comp_const {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [] [] [] {X : Type u_11} [] {g : XF →SL[σ₂₃] G} (hg : ) (f : E →SL[σ₁₂] F) :
def ContinuousLinearMap.compL (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] :
(Fₗ →L[𝕜] Gₗ) →L[𝕜] (E →L[𝕜] Fₗ) →L[𝕜] E →L[𝕜] Gₗ

Composition of continuous linear maps as a continuous bilinear map.

Instances For
theorem ContinuousLinearMap.norm_compL_le (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] :
@[simp]
theorem ContinuousLinearMap.compL_apply (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : Fₗ →L[𝕜] Gₗ) (g : E →L[𝕜] Fₗ) :
↑(↑() f) g =
@[simp]
theorem ContinuousLinearMap.precompR_apply {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) :
∀ (a : E), ↑() a = ↑(ContinuousLinearMap.compL 𝕜 Eₗ Fₗ Gₗ) (L a)
def ContinuousLinearMap.precompR {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) :
E →L[𝕜] (Eₗ →L[𝕜] Fₗ) →L[𝕜] Eₗ →L[𝕜] Gₗ

Apply L(x,-) pointwise to bilinear maps, as a continuous bilinear map

Instances For
def ContinuousLinearMap.precompL {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) :
(Eₗ →L[𝕜] E) →L[𝕜] Fₗ →L[𝕜] Eₗ →L[𝕜] Gₗ

Apply L(-,y) pointwise to bilinear maps, as a continuous bilinear map

Instances For
theorem ContinuousLinearMap.norm_precompR_le {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) :
theorem ContinuousLinearMap.norm_precompL_le {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) :
def ContinuousLinearMap.prodMapL (𝕜 : Type u_1) (M₁ : Type u₁) [NormedSpace 𝕜 M₁] (M₂ : Type u₂) [NormedSpace 𝕜 M₂] (M₃ : Type u₃) [NormedSpace 𝕜 M₃] (M₄ : Type u₄) [NormedSpace 𝕜 M₄] :
(M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄) →L[𝕜] M₁ × M₃ →L[𝕜] M₂ × M₄

ContinuousLinearMap.prodMap as a continuous linear map.

Instances For
@[simp]
theorem ContinuousLinearMap.prodMapL_apply (𝕜 : Type u_1) {M₁ : Type u₁} [NormedSpace 𝕜 M₁] {M₂ : Type u₂} [NormedSpace 𝕜 M₂] {M₃ : Type u₃} [NormedSpace 𝕜 M₃] {M₄ : Type u₄} [NormedSpace 𝕜 M₄] (p : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄)) :
↑(ContinuousLinearMap.prodMapL 𝕜 M₁ M₂ M₃ M₄) p = ContinuousLinearMap.prodMap p.fst p.snd
theorem Continuous.prod_mapL (𝕜 : Type u_1) {M₁ : Type u₁} [NormedSpace 𝕜 M₁] {M₂ : Type u₂} [NormedSpace 𝕜 M₂] {M₃ : Type u₃} [NormedSpace 𝕜 M₃] {M₄ : Type u₄} [NormedSpace 𝕜 M₄] {X : Type u_11} [] {f : XM₁ →L[𝕜] M₂} {g : XM₃ →L[𝕜] M₄} (hf : ) (hg : ) :
theorem Continuous.prod_map_equivL (𝕜 : Type u_1) {M₁ : Type u₁} [NormedSpace 𝕜 M₁] {M₂ : Type u₂} [NormedSpace 𝕜 M₂] {M₃ : Type u₃} [NormedSpace 𝕜 M₃] {M₄ : Type u₄} [NormedSpace 𝕜 M₄] {X : Type u_11} [] {f : XM₁ ≃L[𝕜] M₂} {g : XM₃ ≃L[𝕜] M₄} (hf : Continuous fun x => ↑(f x)) (hg : Continuous fun x => ↑(g x)) :
Continuous fun x => ↑(ContinuousLinearEquiv.prod (f x) (g x))
theorem ContinuousOn.prod_mapL (𝕜 : Type u_1) {M₁ : Type u₁} [NormedSpace 𝕜 M₁] {M₂ : Type u₂} [NormedSpace 𝕜 M₂] {M₃ : Type u₃} [NormedSpace 𝕜 M₃] {M₄ : Type u₄} [NormedSpace 𝕜 M₄] {X : Type u_11} [] {f : XM₁ →L[𝕜] M₂} {g : XM₃ →L[𝕜] M₄} {s : Set X} (hf : ) (hg : ) :
ContinuousOn (fun x => ContinuousLinearMap.prodMap (f x) (g x)) s
theorem ContinuousOn.prod_map_equivL (𝕜 : Type u_1) {M₁ : Type u₁} [NormedSpace 𝕜 M₁] {M₂ : Type u₂} [NormedSpace 𝕜 M₂] {M₃ : Type u₃} [NormedSpace 𝕜 M₃] {M₄ : Type u₄} [NormedSpace 𝕜 M₄] {X : Type u_11} [] {f : XM₁ ≃L[𝕜] M₂} {g : XM₃ ≃L[𝕜] M₄} {s : Set X} (hf : ContinuousOn (fun x => ↑(f x)) s) (hg : ContinuousOn (fun x => ↑(g x)) s) :
ContinuousOn (fun x => ↑(ContinuousLinearEquiv.prod (f x) (g x))) s
def ContinuousLinearMap.mul (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] :
𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜'

Multiplication in a non-unital normed algebra as a continuous bilinear map.

Instances For
@[simp]
theorem ContinuousLinearMap.mul_apply' (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] (x : 𝕜') (y : 𝕜') :
↑(↑() x) y = x * y
@[simp]
theorem ContinuousLinearMap.op_norm_mul_apply_le (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] (x : 𝕜') :
↑() x x
theorem ContinuousLinearMap.op_norm_mul_le (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] :
def NonUnitalAlgHom.Lmul (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] :
𝕜' →ₙₐ[𝕜] 𝕜' →L[𝕜] 𝕜'

Multiplication on the left in a non-unital normed algebra 𝕜' as a non-unital algebra homomorphism into the algebra of continuous linear maps. This is the left regular representation of A acting on itself.

This has more algebraic structure than ContinuousLinearMap.mul, but there is no longer continuity bundled in the first coordinate. An alternative viewpoint is that this upgrades NonUnitalAlgHom.lmul from a homomorphism into linear maps to a homomorphism into continuous linear maps.

Instances For
@[simp]
theorem NonUnitalAlgHom.coe_Lmul {𝕜 : Type u_1} {𝕜' : Type u_11} [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] :
↑() = ↑()
def ContinuousLinearMap.mulLeftRight (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] :
𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜'

Simultaneous left- and right-multiplication in a non-unital normed algebra, considered as a continuous trilinear map. This is akin to its non-continuous version LinearMap.mulLeftRight, but there is a minor difference: LinearMap.mulLeftRight is uncurried.

Instances For
@[simp]
theorem ContinuousLinearMap.mulLeftRight_apply (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] (x : 𝕜') (y : 𝕜') (z : 𝕜') :
↑(↑(↑() x) y) z = x * z * y
theorem ContinuousLinearMap.op_norm_mulLeftRight_apply_apply_le (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] (x : 𝕜') (y : 𝕜') :
↑(↑() x) y x * y
theorem ContinuousLinearMap.op_norm_mulLeftRight_apply_le (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] (x : 𝕜') :
↑() x x
theorem ContinuousLinearMap.op_norm_mulLeftRight_le (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] :
class RegularNormedAlgebra (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] :
• isometry_mul' : Isometry ↑()

The left regular representation of the algebra on itself is an isometry.

This is a mixin class for non-unital normed algebras which states that the left-regular representation of the algebra on itself is isometric. Every unital normed algebra with ‖1‖ = 1 is a regular normed algebra (see NormedAlgebra.instRegularNormedAlgebra). In addition, so is every C⋆-algebra, non-unital included (see CstarRing.instRegularNormedAlgebra), but there are yet other examples. Any algebra with an approximate identity (e.g., $$L^1$$) is also regular.

This is a useful class because it gives rise to a nice norm on the unitization; in particular it is a C⋆-norm when the norm on A is a C⋆-norm.

Instances
instance NormedAlgebra.instRegularNormedAlgebra {𝕜 : Type u_12} {𝕜' : Type u_13} [] [] [] :

Every (unital) normed algebra such that ‖1‖ = 1 is a RegularNormedAlgebra.

theorem ContinuousLinearMap.isometry_mul (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] [] :
Isometry ↑()
@[simp]
theorem ContinuousLinearMap.op_norm_mul_apply (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] [] (x : 𝕜') :
↑() x = x
@[simp]
theorem ContinuousLinearMap.op_nnnorm_mul_apply (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] [] (x : 𝕜') :
def ContinuousLinearMap.mulₗᵢ (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] [] :
𝕜' →ₗᵢ[𝕜] 𝕜' →L[𝕜] 𝕜'

Multiplication in a normed algebra as a linear isometry to the space of continuous linear maps.

Instances For
@[simp]
theorem ContinuousLinearMap.coe_mulₗᵢ (𝕜 : Type u_1) (𝕜' : Type u_11) [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] [] :
↑() = ↑()
def ContinuousLinearMap.lsmul (𝕜 : Type u_1) {E : Type u_4} [] (𝕜' : Type u_11) [] [] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] :
𝕜' →L[𝕜] E →L[𝕜] E

Scalar multiplication as a continuous bilinear map.

Instances For
@[simp]
theorem ContinuousLinearMap.lsmul_apply (𝕜 : Type u_1) {E : Type u_4} [] (𝕜' : Type u_11) [] [] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] (c : 𝕜') (x : E) :
↑(↑() c) x = c x
theorem ContinuousLinearMap.norm_toSpanSingleton (𝕜 : Type u_1) {E : Type u_4} [] (x : E) :
theorem ContinuousLinearMap.op_norm_lsmul_apply_le {𝕜 : Type u_1} {E : Type u_4} [] {𝕜' : Type u_11} [] [] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] (x : 𝕜') :
↑() x x
theorem ContinuousLinearMap.op_norm_lsmul_le {𝕜 : Type u_1} {E : Type u_4} [] {𝕜' : Type u_11} [] [] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] :

The norm of lsmul is at most 1 in any semi-normed group.

@[simp]
theorem ContinuousLinearMap.norm_restrictScalars {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [] [NormedSpace 𝕜 Fₗ] {𝕜' : Type u_11} [] [NormedSpace 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [NormedSpace 𝕜' Fₗ] [IsScalarTower 𝕜' 𝕜 Fₗ] (f : E →L[𝕜] Fₗ) :
def ContinuousLinearMap.restrictScalarsIsometry (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) [] [NormedSpace 𝕜 Fₗ] (𝕜' : Type u_11) [] [NormedSpace 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [NormedSpace 𝕜' Fₗ] [IsScalarTower 𝕜' 𝕜 Fₗ] (𝕜'' : Type u_12) [Ring 𝕜''] [Module 𝕜'' Fₗ] [ContinuousConstSMul 𝕜'' Fₗ] [SMulCommClass 𝕜 𝕜'' Fₗ] [SMulCommClass 𝕜' 𝕜'' Fₗ] :
(E →L[𝕜] Fₗ) →ₗᵢ[𝕜''] E →L[𝕜'] Fₗ

ContinuousLinearMap.restrictScalars as a LinearIsometry.

Instances For
@[simp]
theorem ContinuousLinearMap.coe_restrictScalarsIsometry (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) [] [NormedSpace 𝕜 Fₗ] (𝕜' : Type u_11) [] [NormedSpace 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [NormedSpace 𝕜' Fₗ] [IsScalarTower 𝕜' 𝕜 Fₗ] {𝕜'' : Type u_12} [Ring 𝕜''] [Module 𝕜'' Fₗ] [ContinuousConstSMul 𝕜'' Fₗ] [SMulCommClass 𝕜 𝕜'' Fₗ] [SMulCommClass 𝕜' 𝕜'' Fₗ] :
↑() =
@[simp]
theorem ContinuousLinearMap.restrictScalarsIsometry_toLinearMap (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) [] [NormedSpace 𝕜 Fₗ] (𝕜' : Type u_11) [] [NormedSpace 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [NormedSpace 𝕜' Fₗ] [IsScalarTower 𝕜' 𝕜 Fₗ] {𝕜'' : Type u_12} [Ring 𝕜''] [Module 𝕜'' Fₗ] [ContinuousConstSMul 𝕜'' Fₗ] [SMulCommClass 𝕜 𝕜'' Fₗ] [SMulCommClass 𝕜' 𝕜'' Fₗ] :
().toLinearMap = ContinuousLinearMap.restrictScalarsₗ 𝕜 E Fₗ 𝕜' 𝕜''
def ContinuousLinearMap.restrictScalarsL (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) [] [NormedSpace 𝕜 Fₗ] (𝕜' : Type u_11) [] [NormedSpace 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [NormedSpace 𝕜' Fₗ] [IsScalarTower 𝕜' 𝕜 Fₗ] (𝕜'' : Type u_12) [Ring 𝕜''] [Module 𝕜'' Fₗ] [ContinuousConstSMul 𝕜'' Fₗ] [SMulCommClass 𝕜 𝕜'' Fₗ] [SMulCommClass 𝕜' 𝕜'' Fₗ] :
(E →L[𝕜] Fₗ) →L[𝕜''] E →L[𝕜'] Fₗ

ContinuousLinearMap.restrictScalars as a ContinuousLinearMap.

Instances For
@[simp]
theorem ContinuousLinearMap.coe_restrictScalarsL {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [] [NormedSpace 𝕜 Fₗ] {𝕜' : Type u_11} [] [NormedSpace 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [NormedSpace 𝕜' Fₗ] [IsScalarTower 𝕜' 𝕜 Fₗ] {𝕜'' : Type u_12} [Ring 𝕜''] [Module 𝕜'' Fₗ] [ContinuousConstSMul 𝕜'' Fₗ] [SMulCommClass 𝕜 𝕜'' Fₗ] [SMulCommClass 𝕜' 𝕜'' Fₗ] :
↑(ContinuousLinearMap.restrictScalarsL 𝕜 E Fₗ 𝕜' 𝕜'') = ContinuousLinearMap.restrictScalarsₗ 𝕜 E Fₗ 𝕜' 𝕜''
@[simp]
theorem ContinuousLinearMap.coe_restrict_scalarsL' {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [] [NormedSpace 𝕜 Fₗ] {𝕜' : Type u_11} [] [NormedSpace 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [NormedSpace 𝕜' Fₗ] [IsScalarTower 𝕜' 𝕜 Fₗ] {𝕜'' : Type u_12} [Ring 𝕜''] [Module 𝕜'' Fₗ] [ContinuousConstSMul 𝕜'' Fₗ] [SMulCommClass 𝕜 𝕜'' Fₗ] [SMulCommClass 𝕜' 𝕜'' Fₗ] :
↑(ContinuousLinearMap.restrictScalarsL 𝕜 E Fₗ 𝕜' 𝕜'') =
theorem Submodule.norm_subtypeL_le {𝕜 : Type u_1} {E : Type u_4} [] (K : ) :
theorem ContinuousLinearEquiv.lipschitz {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [] (e : E ≃SL[σ₁₂] F) :
theorem ContinuousLinearEquiv.isBigO_comp {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [] (e : E ≃SL[σ₁₂] F) {α : Type u_11} (f : αE) (l : ) :
(fun x' => e (f x')) =O[l] f
theorem ContinuousLinearEquiv.isBigO_sub {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [] (e : E ≃SL[σ₁₂] F) (l : ) (x : E) :
(fun x' => e (x' - x)) =O[l] fun x' => x' - x
theorem ContinuousLinearEquiv.isBigO_comp_rev {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [] (e : E ≃SL[σ₁₂] F) {α : Type u_11} (f : αE) (l : ) :
f =O[l] fun x' => e (f x')
theorem ContinuousLinearEquiv.isBigO_sub_rev {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [] (e : E ≃SL[σ₁₂] F) (l : ) (x : E) :
(fun x' => x' - x) =O[l] fun x' => e (x' - x)
def ContinuousLinearMap.bilinearComp {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} {E' : Type u_11} {F' : Type u_12} {𝕜₁' : Type u_13} {𝕜₂' : Type u_14} [] [] [NormedSpace 𝕜₁' E'] [NormedSpace 𝕜₂' F'] {σ₁' : 𝕜₁' →+* 𝕜} {σ₁₃' : 𝕜₁' →+* 𝕜₃} {σ₂' : 𝕜₂' →+* 𝕜₂} {σ₂₃' : 𝕜₂' →+* 𝕜₃} [RingHomCompTriple σ₁' σ₁₃ σ₁₃'] [RingHomCompTriple σ₂' σ₂₃ σ₂₃'] [] [RingHomIsometric σ₁₃'] [RingHomIsometric σ₂₃'] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (gE : E' →SL[σ₁'] E) (gF : F' →SL[σ₂'] F) :
E' →SL[σ₁₃'] F' →SL[σ₂₃'] G

Compose a bilinear map E →SL[σ₁₃] F →SL[σ₂₃] G with two linear maps E' →SL[σ₁'] E and F' →SL[σ₂'] F.

Instances For
@[simp]
theorem ContinuousLinearMap.bilinearComp_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} {E' : Type u_11} {F' : Type u_12} {𝕜₁' : Type u_13} {𝕜₂' : Type u_14} [] [] [NormedSpace 𝕜₁' E'] [NormedSpace 𝕜₂' F'] {σ₁' : 𝕜₁' →+* 𝕜} {σ₁₃' : 𝕜₁' →+* 𝕜₃} {σ₂' : 𝕜₂' →+* 𝕜₂} {σ₂₃' : 𝕜₂' →+* 𝕜₃} [RingHomCompTriple σ₁' σ₁₃ σ₁₃'] [RingHomCompTriple σ₂' σ₂₃ σ₂₃'] [] [RingHomIsometric σ₁₃'] [RingHomIsometric σ₂₃'] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (gE : E' →SL[σ₁'] E) (gF : F' →SL[σ₂'] F) (x : E') (y : F') :
↑(↑() x) y = ↑(f (gE x)) (gF y)
def ContinuousLinearMap.deriv₂ {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) :
E × Fₗ →L[𝕜] E × Fₗ →L[𝕜] Gₗ

Derivative of a continuous bilinear map f : E →L[𝕜] F →L[𝕜] G interpreted as a map E × F → G at point p : E × F evaluated at q : E × F, as a continuous bilinear map.

Instances For
@[simp]
theorem ContinuousLinearMap.coe_deriv₂ {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (p : E × Fₗ) :
↑() = fun q => ↑(f p.fst) q.snd + ↑(f q.fst) p.snd
theorem ContinuousLinearMap.map_add_add {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (x : E) (x' : E) (y : Fₗ) (y' : Fₗ) :
↑(f (x + x')) (y + y') = ↑(f x) y + ↑(↑() (x, y)) (x', y') + ↑(f x') y'
theorem LinearMap.bound_of_shell {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] (f : E →ₛₗ[σ₁₂] F) {ε : } {C : } (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < c) (hf : ∀ (x : E), ε / c xx < εf x C * x) (x : E) :
f x C * x
theorem LinearMap.bound_of_ball_bound {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [] [] [NormedSpace 𝕜 Fₗ] {r : } (r_pos : 0 < r) (c : ) (f : E →ₗ[𝕜] Fₗ) (h : ∀ (z : E), z f z c) :
C, ∀ (z : E), f z C * z

LinearMap.bound_of_ball_bound' is a version of this lemma over a field satisfying IsROrC that produces a concrete bound.

theorem LinearMap.antilipschitz_of_comap_nhds_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [h : ] (f : E →ₛₗ[σ₁₂] F) (hf : Filter.comap (f) (nhds 0) nhds 0) :
K,
theorem ContinuousLinearMap.op_norm_zero_iff {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) [] :
f = 0 f = 0

An operator is zero iff its norm vanishes.

@[simp]
theorem ContinuousLinearMap.norm_id {𝕜 : Type u_1} {E : Type u_4} [] [] :

If a normed space is non-trivial, then the norm of the identity equals 1.

instance ContinuousLinearMap.normOneClass {𝕜 : Type u_1} {E : Type u_4} [] [] :
instance ContinuousLinearMap.toNormedAddCommGroup {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] :

Continuous linear maps themselves form a normed space with respect to the operator norm.

instance ContinuousLinearMap.toNormedRing {𝕜 : Type u_1} {E : Type u_4} [] :
NormedRing (E →L[𝕜] E)

Continuous linear maps form a normed ring with respect to the operator norm.

theorem ContinuousLinearMap.homothety_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [] [] (f : E →SL[σ₁₂] F) {a : } (hf : ∀ (x : E), f x = a * x) :
theorem ContinuousLinearMap.antilipschitz_of_embedding {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [] [] [NormedSpace 𝕜 Fₗ] (f : E →L[𝕜] Fₗ) (hf : ) :
K,

If a continuous linear map is a topology embedding, then it is expands the distances by a positive factor.

@[simp]
theorem ContinuousLinearMap.ofMemClosureImageCoeBounded_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {F : Type u_6} [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {E' : Type u_11} [NormedSpace 𝕜 E'] [] (f : E'F) {s : Set (E' →SL[σ₁₂] F)} (hs : ) (hf : f closure (FunLike.coe '' s)) :
= f
def ContinuousLinearMap.ofMemClosureImageCoeBounded {𝕜 : Type u_1} {𝕜₂ : Type u_2} {F : Type u_6} [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {E' : Type u_11} [NormedSpace 𝕜 E'] [] (f : E'F) {s : Set (E' →SL[σ₁₂] F)} (hs : ) (hf : f closure (FunLike.coe '' s)) :
E' →SL[σ₁₂] F

Construct a bundled continuous (semi)linear map from a map f : E → F and a proof of the fact that it belongs to the closure of the image of a bounded set s : Set (E →SL[σ₁₂] F) under coercion to function. Coercion to function of the result is definitionally equal to f.

Instances For
@[simp]
theorem ContinuousLinearMap.ofTendstoOfBoundedRange_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {F : Type u_6} [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {E' : Type u_11} [NormedSpace 𝕜 E'] [] {α : Type u_12} {l : } [] (f : E'F) (g : αE' →SL[σ₁₂] F) (hf : Filter.Tendsto (fun a x => ↑(g a) x) l (nhds f)) (hg : ) :
↑() = f
def ContinuousLinearMap.ofTendstoOfBoundedRange {𝕜 : Type u_1} {𝕜₂ : Type u_2} {F : Type u_6}