# mathlibdocumentation

analysis.normed_space.enorm

# Extended norm

In this file we define a structure enorm 𝕜 V representing an extended norm (i.e., a norm that can take the value ∞) on a vector space V over a normed field 𝕜. 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 : α → ℝ has a family of L_p extended norms.

We prove some basic inequalities, then define

• emetric_space structure on V corresponding to e : enorm 𝕜 V;
• the subspace of vectors with finite norm, called e.finite_subspace;
• a normed_space structure on this space.

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 (𝕜 : Type u_1) (V : Type u_2) [normed_field 𝕜] [ V] :
Type u_2

Extended norm on a vector space. As in the case of normed spaces, we require only ∥c • x∥ ≤ ∥c∥ * ∥x∥ in the definition, then prove an equality in map_smul.

@[instance]
def enorm.has_coe_to_fun {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] :

Equations
theorem enorm.injective_coe_fn {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] :
function.injective (λ (e : V) (x : V), e x)

@[ext]
theorem enorm.ext {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] {e₁ e₂ : V} :
(∀ (x : V), e₁ x = e₂ x)e₁ = e₂

theorem enorm.ext_iff {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] {e₁ e₂ : V} :
e₁ = e₂ ∀ (x : V), e₁ x = e₂ x

@[simp]
theorem enorm.coe_inj {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] {e₁ e₂ : V} :
e₁ = e₂ e₁ = e₂

@[simp]
theorem enorm.map_smul {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) (c : 𝕜) (x : V) :
e (c x) = ((nnnorm c)) * e x

@[simp]
theorem enorm.map_zero {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) :
e 0 = 0

@[simp]
theorem enorm.eq_zero_iff {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) {x : V} :
e x = 0 x = 0

@[simp]
theorem enorm.map_neg {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) (x : V) :
e (-x) = e x

theorem enorm.map_sub_rev {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) (x y : V) :
e (x - y) = e (y - x)

theorem enorm.map_add_le {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) (x y : V) :
e (x + y) e x + e y

theorem enorm.map_sub_le {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) (x y : V) :
e (x - y) e x + e y

@[instance]
def enorm.partial_order {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] :

Equations
@[instance]
def enorm.has_top {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] :
has_top (enorm 𝕜 V)

The enorm sending each non-zero vector to infinity.

Equations
@[instance]
def enorm.inhabited {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] :
inhabited (enorm 𝕜 V)

Equations
theorem enorm.top_map {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] {x : V} :
x 0 x =

@[instance]
def enorm.semilattice_sup_top {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] :

Equations
@[simp]
theorem enorm.coe_max {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e₁ e₂ : V) :
(e₁ e₂) = λ (x : V), max (e₁ x) (e₂ x)

theorem enorm.max_map {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e₁ e₂ : V) (x : V) :
(e₁ e₂) x = max (e₁ x) (e₂ x)

def enorm.emetric_space {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] :
V

Structure of an emetric_space defined by an extended norm.

Equations
def enorm.finite_subspace {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] :
V V

The subspace of vectors with finite enorm.

Equations
@[instance]
def enorm.metric_space {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) :

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

Equations
theorem enorm.finite_dist_eq {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) (x y : (e.finite_subspace)) :
dist x y = (e (x - y)).to_real

theorem enorm.finite_edist_eq {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) (x y : (e.finite_subspace)) :
y = e (x - y)

@[instance]
def enorm.normed_group {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) :

Normed group instance on e.finite_subspace.

Equations
theorem enorm.finite_norm_eq {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) (x : (e.finite_subspace)) :

@[instance]
def enorm.normed_space {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [ V] (e : V) :

Normed space instance on e.finite_subspace.

Equations