mathlib documentation

analysis.normed_space.enorm

Extended norm #

In this file we define a structure enorm π•œ V representing an extended norm (i.e., a norm that can take the value ∞) on a vector space V over a normed field π•œ. We do not use class for an enorm because the same space can have more than one extended norm. For example, the space of measurable functions f : Ξ± β†’ ℝ has a family of L_p extended norms.

We prove some basic inequalities, then define

The last definition is an instance because the type involves e.

Implementation notes #

We do not define extended normed groups. They can be added to the chain once someone will need them.

Tags #

normed space, extended norm

structure enorm (π•œ : Type u_1) (V : Type u_2) [normed_field π•œ] [add_comm_group V] [module π•œ V] :
Type u_2

Extended norm on a vector space. As in the case of normed spaces, we require only β€–c β€’ xβ€– ≀ β€–cβ€– * β€–xβ€– in the definition, then prove an equality in map_smul.

Instances for enorm
@[protected, instance]
def enorm.has_coe_to_fun {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] :
has_coe_to_fun (enorm π•œ V) (Ξ» (_x : enorm π•œ V), V β†’ ennreal)
Equations
theorem enorm.coe_fn_injective {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] :
@[ext]
theorem enorm.ext {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] {e₁ eβ‚‚ : enorm π•œ V} (h : βˆ€ (x : V), ⇑e₁ x = ⇑eβ‚‚ x) :
e₁ = eβ‚‚
theorem enorm.ext_iff {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] {e₁ eβ‚‚ : enorm π•œ V} :
e₁ = eβ‚‚ ↔ βˆ€ (x : V), ⇑e₁ x = ⇑eβ‚‚ x
@[simp, norm_cast]
theorem enorm.coe_inj {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] {e₁ eβ‚‚ : enorm π•œ V} :
⇑e₁ = ⇑eβ‚‚ ↔ e₁ = eβ‚‚
@[simp]
theorem enorm.map_smul {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) (c : π•œ) (x : V) :
@[simp]
theorem enorm.map_zero {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) :
⇑e 0 = 0
@[simp]
theorem enorm.eq_zero_iff {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) {x : V} :
⇑e x = 0 ↔ x = 0
@[simp]
theorem enorm.map_neg {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) (x : V) :
⇑e (-x) = ⇑e x
theorem enorm.map_sub_rev {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) (x y : V) :
⇑e (x - y) = ⇑e (y - x)
theorem enorm.map_add_le {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) (x y : V) :
⇑e (x + y) ≀ ⇑e x + ⇑e y
theorem enorm.map_sub_le {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) (x y : V) :
⇑e (x - y) ≀ ⇑e x + ⇑e y
@[protected, instance]
def enorm.partial_order {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] :
partial_order (enorm π•œ V)
Equations
@[protected, instance]
noncomputable def enorm.has_top {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] :
has_top (enorm π•œ V)

The enorm sending each non-zero vector to infinity.

Equations
@[protected, instance]
noncomputable def enorm.inhabited {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] :
inhabited (enorm π•œ V)
Equations
theorem enorm.top_map {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] {x : V} (hx : x β‰  0) :
@[protected, instance]
noncomputable def enorm.order_top {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] :
order_top (enorm π•œ V)
Equations
@[protected, instance]
noncomputable def enorm.semilattice_sup {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] :
semilattice_sup (enorm π•œ V)
Equations
@[simp, norm_cast]
theorem enorm.coe_max {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e₁ eβ‚‚ : enorm π•œ V) :
⇑(e₁ βŠ” eβ‚‚) = Ξ» (x : V), linear_order.max (⇑e₁ x) (⇑eβ‚‚ x)
@[norm_cast]
theorem enorm.max_map {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e₁ eβ‚‚ : enorm π•œ V) (x : V) :
⇑(e₁ βŠ” eβ‚‚) x = linear_order.max (⇑e₁ x) (⇑eβ‚‚ x)
@[reducible]
def enorm.emetric_space {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) :

Structure of an emetric_space defined by an extended norm.

Equations
def enorm.finite_subspace {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) :
subspace π•œ V

The subspace of vectors with finite enorm.

Equations
Instances for β†₯enorm.finite_subspace
@[protected, instance]
def enorm.finite_subspace.metric_space {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) :

Metric space structure on e.finite_subspace. We use emetric_space.to_metric_space to ensure that this definition agrees with e.emetric_space.

Equations
theorem enorm.finite_dist_eq {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) (x y : β†₯(e.finite_subspace)) :
theorem enorm.finite_edist_eq {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) (x y : β†₯(e.finite_subspace)) :
theorem enorm.finite_norm_eq {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) (x : β†₯(e.finite_subspace)) :
@[protected, instance]
def enorm.finite_subspace.normed_space {π•œ : Type u_1} {V : Type u_2} [normed_field π•œ] [add_comm_group V] [module π•œ V] (e : enorm π•œ V) :

Normed space instance on e.finite_subspace.

Equations