# mathlib3documentation

analysis.normed_space.operator_norm

# Operator norm on the space of continuous linear maps #

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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 `seminormed_add_comm_group` and we specialize to `normed_add_comm_group` 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 `[ring_hom_isometric 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} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ σ₁₂ E F] (f : 𝓕) (hf : continuous f) {x : E} (hx : x = 0) :

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

theorem semilinear_map_class.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} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] [ σ₁₂ E F] (f : 𝓕) {ε C : } (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < c) (hf : (x : E), ε / c x x < ε f x C * x) {x : E} (hx : x 0) :
theorem semilinear_map_class.bound_of_continuous {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} {𝓕 : Type u_10} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] [ σ₁₂ E F] (f : 𝓕) (hf : continuous f) :
(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 continuous_linear_map.bound {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) :
(C : ), 0 < C (x : E), f x C * x
def linear_isometry.to_span_singleton (𝕜 : Type u_1) (E : Type u_4) [ E] {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`.

Equations
@[simp]
theorem linear_isometry.to_span_singleton_apply {𝕜 : Type u_1} {E : Type u_4} [ E] {v : E} (hv : v = 1) (a : 𝕜) :
a = a v
@[simp]
theorem linear_isometry.coe_to_span_singleton {𝕜 : Type u_1} {E : Type u_4} [ E] {v : E} (hv : v = 1) :
noncomputable def continuous_linear_map.op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) :

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

Equations
@[protected, instance]
noncomputable def continuous_linear_map.has_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} :
has_norm (E →SL[σ₁₂] F)
Equations
theorem continuous_linear_map.norm_def {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) :
f = has_Inf.Inf {c : | 0 c (x : E), f x c * x}
theorem continuous_linear_map.bounds_nonempty {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] {f : E →SL[σ₁₂] F} :
(c : ), c {c : | 0 c (x : E), f x c * x}
theorem continuous_linear_map.bounds_bdd_below {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {f : E →SL[σ₁₂] F} :
bdd_below {c : | 0 c (x : E), f x c * x}
theorem continuous_linear_map.op_norm_le_bound {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ 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 continuous_linear_map.op_norm_le_bound' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) {M : } (hMp : 0 M) (hM : (x : E), x 0 f x M * x) :

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

theorem continuous_linear_map.op_norm_le_of_lipschitz {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {f : E →SL[σ₁₂] F} {K : nnreal} (hf : f) :
theorem continuous_linear_map.op_norm_eq_of_bounds {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ 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 continuous_linear_map.op_norm_neg {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) :
theorem continuous_linear_map.op_norm_nonneg {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) :
theorem continuous_linear_map.le_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) (x : E) :

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

theorem continuous_linear_map.dist_le_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) (x y : E) :
theorem continuous_linear_map.le_op_norm_of_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) {c : } {x : E} (h : x c) :
theorem continuous_linear_map.le_of_op_norm_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) {c : } (h : f c) (x : E) :
theorem continuous_linear_map.ratio_le_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) (x : E) :
theorem continuous_linear_map.unit_le_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) (x : E) :

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

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

The operator norm satisfies the triangle inequality.

theorem continuous_linear_map.op_norm_zero {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] :

The norm of the `0` operator is `0`.

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

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 continuous_linear_map.norm_id_of_nontrivial_seminorm {𝕜 : Type u_1} {E : Type u_4} [ E] (h : (x : E), 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 continuous_linear_map.op_norm_smul_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] {𝕜' : Type u_3} [normed_field 𝕜'] [ F] [ 𝕜' F] (c : 𝕜') (f : E →SL[σ₁₂] F) :
@[protected]
noncomputable def continuous_linear_map.tmp_seminormed_add_comm_group {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] :

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]

Equations
@[protected]
noncomputable def continuous_linear_map.tmp_pseudo_metric_space {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] :

The `pseudo_metric_space` structure on `E →SL[σ₁₂] F` coming from `continuous_linear_map.tmp_seminormed_add_comm_group`. See Note [forgetful inheritance]

Equations
@[protected]
noncomputable def continuous_linear_map.tmp_uniform_space {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] :
uniform_space (E →SL[σ₁₂] F)

The `uniform_space` structure on `E →SL[σ₁₂] F` coming from `continuous_linear_map.tmp_seminormed_add_comm_group`. See Note [forgetful inheritance]

Equations
@[protected]
noncomputable def continuous_linear_map.tmp_topological_space {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] :

The `topological_space` structure on `E →SL[σ₁₂] F` coming from `continuous_linear_map.tmp_seminormed_add_comm_group`. See Note [forgetful inheritance]

Equations
@[protected]
theorem continuous_linear_map.tmp_topological_add_group {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] :
@[protected]
theorem continuous_linear_map.tmp_closed_ball_div_subset {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] {a b : } (ha : 0 < a) (hb : 0 < b) :
(a / b) {f : E →SL[σ₁₂] F | (x : E), x f x
@[protected]
theorem continuous_linear_map.tmp_topology_eq {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] :
@[protected]
theorem continuous_linear_map.tmp_uniform_space_eq {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] :
@[protected, instance]
noncomputable def continuous_linear_map.to_pseudo_metric_space {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] :
Equations
@[protected, instance]
noncomputable def continuous_linear_map.to_seminormed_add_comm_group {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] :

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

Equations
theorem continuous_linear_map.nnnorm_def {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) :
theorem continuous_linear_map.op_nnnorm_le_bound {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (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 continuous_linear_map.op_nnnorm_le_bound' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) (M : nnreal) (hM : (x : E), x‖₊ 0 f x‖₊ M * x‖₊) :

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

theorem continuous_linear_map.op_nnnorm_le_of_unit_nnnorm {E : Type u_4} {F : Type u_6} [ E] [ F] {f : E →L[] F} {C : nnreal} (hf : (x : E), x‖₊ = 1 f 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 continuous_linear_map.op_nnnorm_le_of_lipschitz {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] {f : E →SL[σ₁₂] F} {K : nnreal} (hf : f) :
theorem continuous_linear_map.op_nnnorm_eq_of_bounds {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] {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) :
@[protected, instance]
def continuous_linear_map.to_normed_space {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] {𝕜' : Type u_3} [normed_field 𝕜'] [ F] [ 𝕜' F] :
(E →SL[σ₁₂] F)
Equations
theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ σ₂₃ σ₁₃] [ring_hom_isometric σ₁₂] [ring_hom_isometric σ₂₃] (h : F →SL[σ₂₃] G) (f : E →SL[σ₁₂] F) :

The operator norm is submultiplicative.

theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ σ₂₃ σ₁₃] [ring_hom_isometric σ₁₂] [ring_hom_isometric σ₂₃] (h : F →SL[σ₂₃] G) [ring_hom_isometric σ₁₃] (f : E →SL[σ₁₂] F) :
@[protected, instance]
noncomputable def continuous_linear_map.to_semi_normed_ring {𝕜 : Type u_1} {E : Type u_4} [ E] :

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

Equations
@[protected, instance]
noncomputable def continuous_linear_map.to_normed_algebra {𝕜 : Type u_1} {E : Type u_4} [ E] :
(E →L[𝕜] E)

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

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

continuous linear maps are Lipschitz continuous.

theorem continuous_linear_map.lipschitz_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (x : E) :
(λ (f : E →SL[σ₁₂] F), f x)

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

theorem continuous_linear_map.exists_mul_lt_apply_of_lt_op_nnnorm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) {r : nnreal} (hr : r < f‖₊) :
(x : E), r * x‖₊ < f x‖₊
theorem continuous_linear_map.exists_mul_lt_of_lt_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) {r : } (hr₀ : 0 r) (hr : r < f) :
(x : E), r * x < f x
theorem continuous_linear_map.exists_lt_apply_of_lt_op_nnnorm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {σ₁₂ : 𝕜 →+* 𝕜₂} [ E] [ F] [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) {r : nnreal} (hr : r < f‖₊) :
(x : E), x‖₊ < 1 r < f x‖₊
theorem continuous_linear_map.exists_lt_apply_of_lt_op_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {σ₁₂ : 𝕜 →+* 𝕜₂} [ E] [ F] [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) {r : } (hr : r < f) :
(x : E), x < 1 r < f x
theorem continuous_linear_map.Sup_unit_ball_eq_nnnorm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {σ₁₂ : 𝕜 →+* 𝕜₂} [ E] [ F] [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) :
has_Sup.Sup ((λ (x : E), f x‖₊) '' 1) = f‖₊
theorem continuous_linear_map.Sup_unit_ball_eq_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {σ₁₂ : 𝕜 →+* 𝕜₂} [ E] [ F] [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) :
has_Sup.Sup ((λ (x : E), f x) '' 1) = f
theorem continuous_linear_map.Sup_closed_unit_ball_eq_nnnorm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {σ₁₂ : 𝕜 →+* 𝕜₂} [ E] [ F] [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) :
theorem continuous_linear_map.Sup_closed_unit_ball_eq_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} {σ₁₂ : 𝕜 →+* 𝕜₂} [ E] [ F] [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) :
has_Sup.Sup ((λ (x : E), f x) '' = f
theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₁₃] (f : E →SL[σ₁₂] F) (g : E →SL[σ₁₃] G) (h : (x : E), f x = g x) :
theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) {C : } (h0 : 0 C) (hC : (x : E) (y : F), (f x) y C * x * y) :
theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F) :
@[simp]
theorem continuous_linear_map.op_norm_prod {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Fₗ] [ Gₗ] (f : E →L[𝕜] Fₗ) (g : E →L[𝕜] Gₗ) :
f.prod g = (f, g)
@[simp]
theorem continuous_linear_map.op_nnnorm_prod {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Fₗ] [ Gₗ] (f : E →L[𝕜] Fₗ) (g : E →L[𝕜] Gₗ) :
def continuous_linear_map.prodₗᵢ {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Fₗ] [ Gₗ] (R : Type u_2) [semiring R] [ Fₗ] [ Gₗ] [ Fₗ] [ Gₗ] :
(E →L[𝕜] Fₗ) × (E →L[𝕜] Gₗ) ≃ₗᵢ[R] E →L[𝕜] Fₗ × Gₗ

`continuous_linear_map.prod` as a `linear_isometry_equiv`.

Equations
@[simp]
theorem continuous_linear_map.op_norm_subsingleton {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) [subsingleton E] :
theorem continuous_linear_map.is_O_with_id {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) (l : filter E) :
(λ (x : E), x)
theorem continuous_linear_map.is_O_id {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) (l : filter E) :
f =O[l] λ (x : E), x
theorem continuous_linear_map.is_O_with_comp {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {F : Type u_6} {G : Type u_8} [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} [ring_hom_isometric σ₂₃] {α : Type u_1} (g : F →SL[σ₂₃] G) (f : α F) (l : filter α) :
(λ (x' : α), g (f x')) f
theorem continuous_linear_map.is_O_comp {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {F : Type u_6} {G : Type u_8} [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} [ring_hom_isometric σ₂₃] {α : Type u_1} (g : F →SL[σ₂₃] G) (f : α F) (l : filter α) :
(λ (x' : α), g (f x')) =O[l] f
theorem continuous_linear_map.is_O_with_sub {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) (l : filter E) (x : E) :
(λ (x' : E), f (x' - x)) (λ (x' : E), x' - x)
theorem continuous_linear_map.is_O_sub {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →SL[σ₁₂] F) (l : filter E) (x : E) :
(λ (x' : E), f (x' - x)) =O[l] λ (x' : E), x' - x
theorem linear_isometry.norm_to_continuous_linear_map_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →ₛₗᵢ[σ₁₂] F) :
theorem linear_map.mk_continuous_norm_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ 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 `mk_continuous`, then its norm is bounded by the bound given to the constructor if it is nonnegative.

theorem linear_map.mk_continuous_norm_le' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ 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 `mk_continuous`, then its norm is bounded by the bound or zero if bound is negative.

noncomputable def linear_map.mk_continuous₂ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] (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 `linear_map.mk₂`.

Equations
@[simp]
theorem linear_map.mk_continuous₂_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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : } (hC : (x : E) (y : F), (f x) y C * x * y) (x : E) (y : F) :
((f.mk_continuous₂ C hC) x) y = (f x) y
theorem linear_map.mk_continuous₂_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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : } (hC : (x : E) (y : F), (f x) y C * x * y) :
theorem linear_map.mk_continuous₂_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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : } (h0 : 0 C) (hC : (x : E) (y : F), (f x) y C * x * y) :
hC C
noncomputable def continuous_linear_map.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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] (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 `linear_isometry_equiv`, see `continuous_linear_map.flipL`.

Equations
@[simp]
theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F) :
((f.flip) y) x = (f x) y
@[simp]
theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) :
f.flip.flip = f
@[simp]
theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) :
@[simp]
theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] (f g : E →SL[σ₁₃] F →SL[σ₂₃] G) :
(f + g).flip = f.flip + g.flip
@[simp]
theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] (c : 𝕜₃) (f : E →SL[σ₁₃] F →SL[σ₂₃] G) :
(c f).flip = c f.flip
noncomputable def continuous_linear_map.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) [ E] [ F] [ G] (σ₂₃ : 𝕜₂ →+* 𝕜₃) (σ₁₃ : 𝕜 →+* 𝕜₃) [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] :
(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 `linear_isometry_equiv`. For an unbundled version see `continuous_linear_map.flip`.

Equations
@[simp]
theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] :
σ₂₃ σ₁₃).symm = σ₁₃ σ₂₃
@[simp]
theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] :
σ₂₃ σ₁₃) = continuous_linear_map.flip
noncomputable def continuous_linear_map.flipₗᵢ (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [ E] [ Fₗ] [ 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 `linear_isometry_equiv`. For an unbundled version see `continuous_linear_map.flip`.

Equations
@[simp]
theorem continuous_linear_map.flipₗᵢ_symm {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Fₗ] [ Gₗ] :
Fₗ Gₗ).symm = Gₗ
@[simp]
theorem continuous_linear_map.coe_flipₗᵢ {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Fₗ] [ Gₗ] :
noncomputable def continuous_linear_map.apply' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} (F : Type u_6) [ E] [ F] (σ₁₂ : 𝕜 →+* 𝕜₂) [ring_hom_isometric σ₁₂] :
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 `linear_map.applyₗ`.

Equations
@[simp]
theorem continuous_linear_map.apply_apply' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (v : E) (f : E →SL[σ₁₂] F) :
( σ₁₂) v) f = f v
noncomputable def continuous_linear_map.apply (𝕜 : Type u_1) {E : Type u_4} (Fₗ : Type u_7) [ E] [ 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 `linear_map.applyₗ`.

Equations
@[simp]
theorem continuous_linear_map.apply_apply {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [ E] [ Fₗ] (v : E) (f : E →L[𝕜] Fₗ) :
( v) f = f v
noncomputable def continuous_linear_map.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) [ E] [ F] [ G] (σ₁₂ : 𝕜 →+* 𝕜₂) (σ₂₃ : 𝕜₂ →+* 𝕜₃) {σ₁₃ : 𝕜 →+* 𝕜₃} [ σ₂₃ σ₁₃] [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] [ring_hom_isometric σ₁₂] :
(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.

Equations
theorem continuous_linear_map.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) [ E] [ F] [ G] (σ₁₂ : 𝕜 →+* 𝕜₂) (σ₂₃ : 𝕜₂ →+* 𝕜₃) {σ₁₃ : 𝕜 →+* 𝕜₃} [ σ₂₃ σ₁₃] [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] [ring_hom_isometric σ₁₂] :
σ₁₂ σ₂₃ 1
@[simp]
theorem continuous_linear_map.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} [ E] [ F] [ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ σ₂₃ σ₁₃] [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] [ring_hom_isometric σ₁₂] (f : F →SL[σ₂₃] G) (g : E →SL[σ₁₂] F) :
( σ₁₂ σ₂₃) f) g = f.comp 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} [ E] [ F] [ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ σ₂₃ σ₁₃] [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] [ring_hom_isometric σ₁₂] {X : Type u_5} {f : X (E →SL[σ₁₂] F)} (hf : continuous f) (g : F →SL[σ₂₃] G) :
continuous (λ (x : X), g.comp (f x))
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} [ E] [ F] [ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [ σ₂₃ σ₁₃] [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃] [ring_hom_isometric σ₁₂] {X : Type u_5} {g : X (F →SL[σ₂₃] G)} (hg : continuous g) (f : E →SL[σ₁₂] F) :
continuous (λ (x : X), (g x).comp f)
noncomputable def continuous_linear_map.compL (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [ E] [ Fₗ] [ 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.

Equations
theorem continuous_linear_map.norm_compL_le (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [ E] [ Fₗ] [ Gₗ] :
Gₗ 1
@[simp]
theorem continuous_linear_map.compL_apply (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [ E] [ Fₗ] [ Gₗ] (f : Fₗ →L[𝕜] Gₗ) (g : E →L[𝕜] Fₗ) :
( Fₗ Gₗ) f) g = f.comp g
noncomputable def continuous_linear_map.precompR {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Eₗ] [ Fₗ] [ 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

Equations
@[simp]
theorem continuous_linear_map.precompR_apply {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Eₗ] [ Fₗ] [ Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (ᾰ : E) :
= Fₗ Gₗ) (L ᾰ)
noncomputable def continuous_linear_map.precompL {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Eₗ] [ Fₗ] [ 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

Equations
theorem continuous_linear_map.norm_precompR_le {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Eₗ] [ Fₗ] [ Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) :
theorem continuous_linear_map.norm_precompL_le {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Eₗ] [ Fₗ] [ Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) :
noncomputable def continuous_linear_map.prod_mapL (𝕜 : Type u_1) (M₁ : Type u₁) [ M₁] (M₂ : Type u₂) [ M₂] (M₃ : Type u₃) [ M₃] (M₄ : Type u₄) [ M₄] :
(M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄) →L[𝕜] M₁ × M₃ →L[𝕜] M₂ × M₄

`continuous_linear_map.prod_map` as a continuous linear map.

Equations
@[simp]
theorem continuous_linear_map.prod_mapL_apply (𝕜 : Type u_1) {M₁ : Type u₁} [ M₁] {M₂ : Type u₂} [ M₂] {M₃ : Type u₃} [ M₃] {M₄ : Type u₄} [ M₄] (p : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄)) :
M₂ M₃ M₄) p = p.fst.prod_map p.snd
theorem continuous.prod_mapL (𝕜 : Type u_1) {M₁ : Type u₁} [ M₁] {M₂ : Type u₂} [ M₂] {M₃ : Type u₃} [ M₃] {M₄ : Type u₄} [ M₄] {X : Type u_11} {f : X (M₁ →L[𝕜] M₂)} {g : X (M₃ →L[𝕜] M₄)} (hf : continuous f) (hg : continuous g) :
continuous (λ (x : X), (f x).prod_map (g x))
theorem continuous.prod_map_equivL (𝕜 : Type u_1) {M₁ : Type u₁} [ M₁] {M₂ : Type u₂} [ M₂] {M₃ : Type u₃} [ M₃] {M₄ : Type u₄} [ M₄] {X : Type u_11} {f : X (M₁ ≃L[𝕜] M₂)} {g : X (M₃ ≃L[𝕜] M₄)} (hf : continuous (λ (x : X), (f x))) (hg : continuous (λ (x : X), (g x))) :
continuous (λ (x : X), ((f x).prod (g x)))
theorem continuous_on.prod_mapL (𝕜 : Type u_1) {M₁ : Type u₁} [ M₁] {M₂ : Type u₂} [ M₂] {M₃ : Type u₃} [ M₃] {M₄ : Type u₄} [ M₄] {X : Type u_11} {f : X (M₁ →L[𝕜] M₂)} {g : X (M₃ →L[𝕜] M₄)} {s : set X} (hf : s) (hg : s) :
continuous_on (λ (x : X), (f x).prod_map (g x)) s
theorem continuous_on.prod_map_equivL (𝕜 : Type u_1) {M₁ : Type u₁} [ M₁] {M₂ : Type u₂} [ M₂] {M₃ : Type u₃} [ M₃] {M₄ : Type u₄} [ M₄] {X : Type u_11} {f : X (M₁ ≃L[𝕜] M₂)} {g : X (M₃ ≃L[𝕜] M₄)} {s : set X} (hf : continuous_on (λ (x : X), (f x)) s) (hg : continuous_on (λ (x : X), (g x)) s) :
continuous_on (λ (x : X), ((f x).prod (g x))) s
noncomputable def continuous_linear_map.mul (𝕜 : Type u_1) (𝕜' : Type u_11) [ 𝕜'] [ 𝕜' 𝕜'] [ 𝕜' 𝕜'] :
𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜'

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

Equations
@[simp]
theorem continuous_linear_map.mul_apply' (𝕜 : Type u_1) (𝕜' : Type u_11) [ 𝕜'] [ 𝕜' 𝕜'] [ 𝕜' 𝕜'] (x y : 𝕜') :
( x) y = x * y
@[simp]
theorem continuous_linear_map.op_norm_mul_apply_le (𝕜 : Type u_1) (𝕜' : Type u_11) [ 𝕜'] [ 𝕜' 𝕜'] [ 𝕜' 𝕜'] (x : 𝕜') :
theorem continuous_linear_map.op_norm_mul_le (𝕜 : Type u_1) (𝕜' : Type u_11) [ 𝕜'] [ 𝕜' 𝕜'] [ 𝕜' 𝕜'] :
noncomputable def continuous_linear_map.mul_left_right (𝕜 : Type u_1) (𝕜' : Type u_11) [ 𝕜'] [ 𝕜' 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 `linear_map.mul_left_right`, but there is a minor difference: `linear_map.mul_left_right` is uncurried.

Equations
@[simp]
theorem continuous_linear_map.mul_left_right_apply (𝕜 : Type u_1) (𝕜' : Type u_11) [ 𝕜'] [ 𝕜' 𝕜'] [ 𝕜' 𝕜'] (x y z : 𝕜') :
( x) y) z = x * z * y
theorem continuous_linear_map.op_norm_mul_left_right_apply_apply_le (𝕜 : Type u_1) (𝕜' : Type u_11) [ 𝕜'] [ 𝕜' 𝕜'] [ 𝕜' 𝕜'] (x y : 𝕜') :
theorem continuous_linear_map.op_norm_mul_left_right_apply_le (𝕜 : Type u_1) (𝕜' : Type u_11) [ 𝕜'] [ 𝕜' 𝕜'] [ 𝕜' 𝕜'] (x : 𝕜') :
theorem continuous_linear_map.op_norm_mul_left_right_le (𝕜 : Type u_1) (𝕜' : Type u_11) [ 𝕜'] [ 𝕜' 𝕜'] [ 𝕜' 𝕜'] :
noncomputable def continuous_linear_map.mulₗᵢ (𝕜 : Type u_1) (𝕜' : Type u_11) [semi_normed_ring 𝕜'] [ 𝕜'] [norm_one_class 𝕜'] :
𝕜' →ₗᵢ[𝕜] 𝕜' →L[𝕜] 𝕜'

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

Equations
@[simp]
theorem continuous_linear_map.coe_mulₗᵢ (𝕜 : Type u_1) (𝕜' : Type u_11) [semi_normed_ring 𝕜'] [ 𝕜'] [norm_one_class 𝕜'] :
@[simp]
theorem continuous_linear_map.op_norm_mul_apply (𝕜 : Type u_1) (𝕜' : Type u_11) [semi_normed_ring 𝕜'] [ 𝕜'] [norm_one_class 𝕜'] (x : 𝕜') :
noncomputable def continuous_linear_map.lsmul (𝕜 : Type u_1) {E : Type u_4} [ E] (𝕜' : Type u_11) [normed_field 𝕜'] [ 𝕜'] [ E] [ 𝕜' E] :
𝕜' →L[𝕜] E →L[𝕜] E

Scalar multiplication as a continuous bilinear map.

Equations
• = _
@[simp]
theorem continuous_linear_map.lsmul_apply (𝕜 : Type u_1) {E : Type u_4} [ E] (𝕜' : Type u_11) [normed_field 𝕜'] [ 𝕜'] [ E] [ 𝕜' E] (c : 𝕜') (x : E) :
( c) x = c x
theorem continuous_linear_map.norm_to_span_singleton (𝕜 : Type u_1) {E : Type u_4} [ E] (x : E) :
theorem continuous_linear_map.op_norm_lsmul_apply_le {𝕜 : Type u_1} {E : Type u_4} [ E] {𝕜' : Type u_11} [normed_field 𝕜'] [ 𝕜'] [ E] [ 𝕜' E] (x : 𝕜') :
theorem continuous_linear_map.op_norm_lsmul_le {𝕜 : Type u_1} {E : Type u_4} [ E] {𝕜' : Type u_11} [normed_field 𝕜'] [ 𝕜'] [ E] [ 𝕜' E] :

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

@[simp]
theorem continuous_linear_map.norm_restrict_scalars {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [ E] [ Fₗ] {𝕜' : Type u_11} [ 𝕜] [ E] [ 𝕜 E] [ Fₗ] [ 𝕜 Fₗ] (f : E →L[𝕜] Fₗ) :
def continuous_linear_map.restrict_scalars_isometry (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) [ E] [ Fₗ] (𝕜' : Type u_11) [ 𝕜] [ E] [ 𝕜 E] [ Fₗ] [ 𝕜 Fₗ] (𝕜'' : Type u_12) [ring 𝕜''] [module 𝕜'' Fₗ] [ Fₗ] [ 𝕜'' Fₗ] [ 𝕜'' Fₗ] :
(E →L[𝕜] Fₗ) →ₗᵢ[𝕜''] E →L[𝕜'] Fₗ

`continuous_linear_map.restrict_scalars` as a `linear_isometry`.

Equations
• 𝕜'' =
@[simp]
theorem continuous_linear_map.coe_restrict_scalars_isometry {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [ E] [ Fₗ] {𝕜' : Type u_11} [ 𝕜] [ E] [ 𝕜 E] [ Fₗ] [ 𝕜 Fₗ] {𝕜'' : Type u_12} [ring 𝕜''] [module 𝕜'' Fₗ] [ Fₗ] [ 𝕜'' Fₗ] [ 𝕜'' Fₗ] :
@[simp]
theorem continuous_linear_map.restrict_scalars_isometry_to_linear_map {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [ E] [ Fₗ] {𝕜' : Type u_11} [ 𝕜] [ E] [ 𝕜 E] [ Fₗ] [ 𝕜 Fₗ] {𝕜'' : Type u_12} [ring 𝕜''] [module 𝕜'' Fₗ] [ Fₗ] [ 𝕜'' Fₗ] [ 𝕜'' Fₗ] :
𝕜'').to_linear_map = 𝕜''
noncomputable def continuous_linear_map.restrict_scalarsL (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) [ E] [ Fₗ] (𝕜' : Type u_11) [ 𝕜] [ E] [ 𝕜 E] [ Fₗ] [ 𝕜 Fₗ] (𝕜'' : Type u_12) [ring 𝕜''] [module 𝕜'' Fₗ] [ Fₗ] [ 𝕜'' Fₗ] [ 𝕜'' Fₗ] :
(E →L[𝕜] Fₗ) →L[𝕜''] E →L[𝕜'] Fₗ

`continuous_linear_map.restrict_scalars` as a `continuous_linear_map`.

Equations
• 𝕜'' =
@[simp]
theorem continuous_linear_map.coe_restrict_scalarsL {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [ E] [ Fₗ] {𝕜' : Type u_11} [ 𝕜] [ E] [ 𝕜 E] [ Fₗ] [ 𝕜 Fₗ] {𝕜'' : Type u_12} [ring 𝕜''] [module 𝕜'' Fₗ] [ Fₗ] [ 𝕜'' Fₗ] [ 𝕜'' Fₗ] :
𝕜' 𝕜'') = 𝕜''
@[simp]
theorem continuous_linear_map.coe_restrict_scalarsL' {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [ E] [ Fₗ] {𝕜' : Type u_11} [ 𝕜] [ E] [ 𝕜 E] [ Fₗ] [ 𝕜 Fₗ] {𝕜'' : Type u_12} [ring 𝕜''] [module 𝕜'' Fₗ] [ Fₗ] [ 𝕜'' Fₗ] [ 𝕜'' Fₗ] :
theorem submodule.norm_subtypeL_le {𝕜 : Type u_1} {E : Type u_4} [ E] (K : E) :
@[protected]
theorem continuous_linear_equiv.lipschitz {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [ σ₂₁] [ σ₁₂] [ring_hom_isometric σ₁₂] (e : E ≃SL[σ₁₂] F) :
theorem continuous_linear_equiv.is_O_comp {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [ σ₂₁] [ σ₁₂] [ring_hom_isometric σ₁₂] (e : E ≃SL[σ₁₂] F) {α : Type u_3} (f : α E) (l : filter α) :
(λ (x' : α), e (f x')) =O[l] f
theorem continuous_linear_equiv.is_O_sub {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [ σ₂₁] [ σ₁₂] [ring_hom_isometric σ₁₂] (e : E ≃SL[σ₁₂] F) (l : filter E) (x : E) :
(λ (x' : E), e (x' - x)) =O[l] λ (x' : E), x' - x
theorem continuous_linear_equiv.is_O_comp_rev {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [ σ₂₁] [ σ₁₂] [ring_hom_isometric σ₂₁] (e : E ≃SL[σ₁₂] F) {α : Type u_3} (f : α E) (l : filter α) :
f =O[l] λ (x' : α), e (f x')
theorem continuous_linear_equiv.is_O_sub_rev {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [ σ₂₁] [ σ₁₂] [ring_hom_isometric σ₂₁] (e : E ≃SL[σ₁₂] F) (l : filter E) (x : E) :
(λ (x' : E), x' - x) =O[l] λ (x' : E), e (x' - x)
noncomputable def continuous_linear_map.bilinear_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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} {E' : Type u_11} {F' : Type u_12} {𝕜₁' : Type u_13} {𝕜₂' : Type u_14} [normed_space 𝕜₁' E'] [normed_space 𝕜₂' F'] {σ₁' : 𝕜₁' →+* 𝕜} {σ₁₃' : 𝕜₁' →+* 𝕜₃} {σ₂' : 𝕜₂' →+* 𝕜₂} {σ₂₃' : 𝕜₂' →+* 𝕜₃} [ σ₁₃ σ₁₃'] [ σ₂₃ σ₂₃'] [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃'] [ring_hom_isometric 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`.

Equations
@[simp]
theorem continuous_linear_map.bilinear_comp_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} [ E] [ F] [ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} {E' : Type u_11} {F' : Type u_12} {𝕜₁' : Type u_13} {𝕜₂' : Type u_14} [normed_space 𝕜₁' E'] [normed_space 𝕜₂' F'] {σ₁' : 𝕜₁' →+* 𝕜} {σ₁₃' : 𝕜₁' →+* 𝕜₃} {σ₂' : 𝕜₂' →+* 𝕜₂} {σ₂₃' : 𝕜₂' →+* 𝕜₃} [ σ₁₃ σ₁₃'] [ σ₂₃ σ₂₃'] [ring_hom_isometric σ₂₃] [ring_hom_isometric σ₁₃'] [ring_hom_isometric σ₂₃'] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (gE : E' →SL[σ₁'] E) (gF : F' →SL[σ₂'] F) (x : E') (y : F') :
((f.bilinear_comp gE gF) x) y = (f (gE x)) (gF y)
noncomputable def continuous_linear_map.deriv₂ {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Fₗ] [ 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.

Equations
@[simp]
theorem continuous_linear_map.coe_deriv₂ {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Fₗ] [ Gₗ] (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (p : E × Fₗ) :
((f.deriv₂) p) = λ (q : E × Fₗ), (f p.fst) q.snd + (f q.fst) p.snd
theorem continuous_linear_map.map_add_add {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [ E] [ Fₗ] [ Gₗ] (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (x x' : E) (y y' : Fₗ) :
(f (x + x')) (y + y') = (f x) y + ((f.deriv₂) (x, y)) (x', y') + (f x') y'
theorem linear_map.bound_of_shell {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) {ε C : } (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < c) (hf : (x : E), ε / c x x < ε f x C * x) (x : E) :
theorem linear_map.bound_of_ball_bound {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [ E] [ Fₗ] {r : } (r_pos : 0 < r) (c : ) (f : E →ₗ[𝕜] Fₗ) (h : (z : E), z r f z c) :
(C : ), (z : E), f z C * z

`linear_map.bound_of_ball_bound'` is a version of this lemma over a field satisfying `is_R_or_C` that produces a concrete bound.

theorem linear_map.antilipschitz_of_comap_nhds_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [h : ring_hom_isometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) (hf : (nhds 0) nhds 0) :
(K : nnreal),
theorem continuous_linear_map.op_norm_zero_iff {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) [ring_hom_isometric σ₁₂] :
f = 0 f = 0

An operator is zero iff its norm vanishes.

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

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

@[protected, instance]
def continuous_linear_map.norm_one_class {𝕜 : Type u_1} {E : Type u_4} [ E] [nontrivial E] :
@[protected, instance]
noncomputable def continuous_linear_map.to_normed_add_comm_group {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] :

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

Equations
@[protected, instance]
noncomputable def continuous_linear_map.to_normed_ring {𝕜 : Type u_1} {E : Type u_4} [ E] :

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

Equations
theorem continuous_linear_map.homothety_norm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [ E] [ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [ring_hom_isometric σ₁₂] [nontrivial E] (f : E →SL[σ₁₂] F) {a : } (hf : (x : E), f x = a