# mathlibdocumentation

analysis.normed_space.dual

# The topological dual of a normed space #

In this file we define the topological dual normed_space.dual of a normed space, and the continuous linear map normed_space.inclusion_in_double_dual from a normed space into its double dual.

For base field 𝕜 = ℝ or 𝕜 = ℂ, this map is actually an isometric embedding; we provide a version normed_space.inclusion_in_double_dual_li of the map which is of type a bundled linear isometric embedding, E →ₗᵢ[𝕜] (dual 𝕜 (dual 𝕜 E)).

Since a lot of elementary properties don't require eq_of_dist_eq_zero we start setting up the theory for semi_normed_group and we specialize to normed_group when needed.

## Main definitions #

• inclusion_in_double_dual and inclusion_in_double_dual_li are the inclusion of a normed space in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
• polar 𝕜 s is the subset of dual 𝕜 E consisting of those functionals x' for which ∥x' z∥ ≤ 1 for every z ∈ s.

## Tags #

dual

@[protected, instance]
def normed_space.dual.inhabited (𝕜 : Type u_1) (E : Type u_2) [ E] :
def normed_space.dual (𝕜 : Type u_1) (E : Type u_2) [ E] :
Type (max u_2 u_1)

The topological dual of a seminormed space E.

Equations
Instances for normed_space.dual
@[protected, instance]
noncomputable def normed_space.dual.semi_normed_group (𝕜 : Type u_1) (E : Type u_2) [ E] :
@[protected, instance]
def normed_space.dual.normed_space (𝕜 : Type u_1) (E : Type u_2) [ E] :
E)
@[protected, instance]
def normed_space.add_monoid_hom_class (𝕜 : Type u_1) (E : Type u_2) [ E] :
𝕜
Equations
@[protected, instance]
def normed_space.dual.has_coe_to_fun (𝕜 : Type u_1) (E : Type u_2) [ E] :
(λ (_x : , E → 𝕜)
Equations
@[protected, instance]
noncomputable def normed_space.dual.normed_group (𝕜 : Type u_1) (F : Type u_3) [normed_group F] [ F] :
Equations
@[protected, instance]
def normed_space.dual.finite_dimensional (𝕜 : Type u_1) (E : Type u_2) [ E] [ E] :
noncomputable def normed_space.inclusion_in_double_dual (𝕜 : Type u_1) (E : Type u_2) [ E] :
E →L[𝕜]

The inclusion of a normed space in its double (topological) dual, considered as a bounded linear map.

Equations
@[simp]
theorem normed_space.dual_def (𝕜 : Type u_1) (E : Type u_2) [ E] (x : E) (f : E) :
f = f x
theorem normed_space.inclusion_in_double_dual_norm_eq (𝕜 : Type u_1) (E : Type u_2) [ E] :
theorem normed_space.inclusion_in_double_dual_norm_le (𝕜 : Type u_1) (E : Type u_2) [ E] :
theorem normed_space.double_dual_bound (𝕜 : Type u_1) (E : Type u_2) [ E] (x : E) :
def normed_space.dual_pairing (𝕜 : Type u_1) (E : Type u_2) [ E] :
→ₗ[𝕜] E →ₗ[𝕜] 𝕜

The dual pairing as a bilinear form.

Equations
@[simp]
theorem normed_space.dual_pairing_apply (𝕜 : Type u_1) (E : Type u_2) [ E] {v : E} {x : E} :
( v) x = v x
theorem normed_space.dual_pairing_separating_left (𝕜 : Type u_1) (E : Type u_2) [ E] :
theorem normed_space.norm_le_dual_bound (𝕜 : Type v) [is_R_or_C 𝕜] {E : Type u} [normed_group E] [ E] (x : E) {M : } (hMp : 0 M) (hM : ∀ (f : , f x M * f) :

If one controls the norm of every f x, then one controls the norm of x. Compare continuous_linear_map.op_norm_le_bound.

theorem normed_space.eq_zero_of_forall_dual_eq_zero (𝕜 : Type v) [is_R_or_C 𝕜] {E : Type u} [normed_group E] [ E] {x : E} (h : ∀ (f : , f x = 0) :
x = 0
theorem normed_space.eq_zero_iff_forall_dual_eq_zero (𝕜 : Type v) [is_R_or_C 𝕜] {E : Type u} [normed_group E] [ E] (x : E) :
x = 0 ∀ (g : , g x = 0
theorem normed_space.eq_iff_forall_dual_eq (𝕜 : Type v) [is_R_or_C 𝕜] {E : Type u} [normed_group E] [ E] {x y : E} :
x = y ∀ (g : , g x = g y
noncomputable def normed_space.inclusion_in_double_dual_li (𝕜 : Type v) [is_R_or_C 𝕜] {E : Type u} [normed_group E] [ E] :

The inclusion of a normed space in its double dual is an isometry onto its image.

Equations
def normed_space.polar (𝕜 : Type u_1) {E : Type u_2} [ E] :
set Eset E)

Given a subset s in a normed space E (over a field 𝕜), the polar polar 𝕜 s is the subset of dual 𝕜 E consisting of those functionals which evaluate to something of norm at most one at all points z ∈ s.

Equations
theorem normed_space.mem_polar_iff (𝕜 : Type u_1) {E : Type u_2} [ E] {x' : E} (s : set E) :
x' ∀ (z : E), z sx' z 1
@[simp]
theorem normed_space.polar_univ (𝕜 : Type u_1) {E : Type u_2} [ E] :
theorem normed_space.is_closed_polar (𝕜 : Type u_1) {E : Type u_2} [ E] (s : set E) :
@[simp]
theorem normed_space.polar_closure (𝕜 : Type u_1) {E : Type u_2} [ E] (s : set E) :
=
theorem normed_space.smul_mem_polar {𝕜 : Type u_1} {E : Type u_2} [ E] {s : set E} {x' : E} {c : 𝕜} (hc : ∀ (z : E), z sx' z c) :

If x' is a dual element such that the norms ∥x' z∥ are bounded for z ∈ s, then a small scalar multiple of x' is in polar 𝕜 s.

theorem normed_space.polar_ball_subset_closed_ball_div {𝕜 : Type u_1} {E : Type u_2} [ E] {c : 𝕜} (hc : 1 < c) {r : } (hr : 0 < r) :
r) (c / r)
theorem normed_space.closed_ball_inv_subset_polar_closed_ball (𝕜 : Type u_1) {E : Type u_2} [ E] {r : } :
theorem normed_space.polar_closed_ball {𝕜 : Type u_1} [is_R_or_C 𝕜] {E : Type u_2} [normed_group E] [ E] {r : } (hr : 0 < r) :

The polar of closed ball in a normed space E is the closed ball of the dual with inverse radius.

theorem normed_space.bounded_polar_of_mem_nhds_zero (𝕜 : Type u_1) {E : Type u_2} [ E] {s : set E} (s_nhd : s nhds 0) :

Given a neighborhood s of the origin in a normed space E, the dual norms of all elements of the polar polar 𝕜 s are bounded by a constant.