mathlib documentation

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 π•œ = ℝ or π•œ = β„‚, 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 β†’β‚—α΅’[π•œ] (dual π•œ (dual π•œ E)).

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

Main definitions #

Tags #

dual

@[protected, instance]
def normed_space.dual.inhabited (π•œ : Type u_1) [nontrivially_normed_field π•œ] (E : Type u_2) [seminormed_add_comm_group E] [normed_space π•œ E] :
@[protected, instance]
@[protected, instance]
def normed_space.dual.normed_space (π•œ : Type u_1) [nontrivially_normed_field π•œ] (E : Type u_2) [seminormed_add_comm_group E] [normed_space π•œ E] :
normed_space π•œ (normed_space.dual π•œ E)
@[protected, instance]
def normed_space.dual.has_coe_to_fun (π•œ : Type u_1) [nontrivially_normed_field π•œ] (E : Type u_2) [seminormed_add_comm_group E] [normed_space π•œ E] :
has_coe_to_fun (normed_space.dual π•œ E) (Ξ» (_x : normed_space.dual π•œ E), E β†’ π•œ)
Equations
@[protected, instance]
def normed_space.dual.finite_dimensional (π•œ : Type u_1) [nontrivially_normed_field π•œ] (E : Type u_2) [seminormed_add_comm_group E] [normed_space π•œ E] [finite_dimensional π•œ E] :
noncomputable def normed_space.inclusion_in_double_dual (π•œ : Type u_1) [nontrivially_normed_field π•œ] (E : Type u_2) [seminormed_add_comm_group E] [normed_space π•œ E] :
E β†’L[π•œ] normed_space.dual π•œ (normed_space.dual π•œ E)

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

Equations
@[simp]
theorem normed_space.dual_def (π•œ : Type u_1) [nontrivially_normed_field π•œ] (E : Type u_2) [seminormed_add_comm_group E] [normed_space π•œ E] (x : E) (f : normed_space.dual π•œ E) :
def normed_space.dual_pairing (π•œ : Type u_1) [nontrivially_normed_field π•œ] (E : Type u_2) [seminormed_add_comm_group E] [normed_space π•œ E] :
normed_space.dual π•œ E β†’β‚—[π•œ] E β†’β‚—[π•œ] π•œ

The dual pairing as a bilinear form.

Equations
@[simp]
theorem normed_space.dual_pairing_apply (π•œ : Type u_1) [nontrivially_normed_field π•œ] (E : Type u_2) [seminormed_add_comm_group E] [normed_space π•œ E] {v : normed_space.dual π•œ E} {x : E} :
theorem normed_space.norm_le_dual_bound (π•œ : Type v) [is_R_or_C π•œ] {E : Type u} [normed_add_comm_group E] [normed_space π•œ E] (x : E) {M : ℝ} (hMp : 0 ≀ M) (hM : βˆ€ (f : normed_space.dual π•œ E), ‖⇑f xβ€– ≀ M * β€–fβ€–) :

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 (π•œ : Type v) [is_R_or_C π•œ] {E : Type u} [normed_add_comm_group E] [normed_space π•œ E] {x : E} (h : βˆ€ (f : normed_space.dual π•œ E), ⇑f x = 0) :
x = 0
theorem normed_space.eq_zero_iff_forall_dual_eq_zero (π•œ : Type v) [is_R_or_C π•œ] {E : Type u} [normed_add_comm_group E] [normed_space π•œ E] (x : E) :
x = 0 ↔ βˆ€ (g : normed_space.dual π•œ E), ⇑g x = 0
theorem normed_space.eq_iff_forall_dual_eq (π•œ : Type v) [is_R_or_C π•œ] {E : Type u} [normed_add_comm_group E] [normed_space π•œ E] {x y : E} :
x = y ↔ βˆ€ (g : normed_space.dual π•œ E), ⇑g x = ⇑g y

See also geometric_hahn_banach_point_point.

noncomputable def normed_space.inclusion_in_double_dual_li (π•œ : Type v) [is_R_or_C π•œ] {E : Type u} [normed_add_comm_group E] [normed_space π•œ E] :

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

Equations
def normed_space.polar (π•œ : Type u_1) [nontrivially_normed_field π•œ] {E : Type u_2} [seminormed_add_comm_group E] [normed_space π•œ E] :

Given a subset s in a normed space E (over a field π•œ), the polar polar π•œ s is the subset of dual π•œ 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 (π•œ : Type u_1) [nontrivially_normed_field π•œ] {E : Type u_2} [seminormed_add_comm_group E] [normed_space π•œ E] {x' : normed_space.dual π•œ E} (s : set E) :
@[simp]
theorem normed_space.polar_univ (π•œ : Type u_1) [nontrivially_normed_field π•œ] {E : Type u_2} [seminormed_add_comm_group E] [normed_space π•œ E] :
theorem normed_space.is_closed_polar (π•œ : Type u_1) [nontrivially_normed_field π•œ] {E : Type u_2} [seminormed_add_comm_group E] [normed_space π•œ E] (s : set E) :
@[simp]
theorem normed_space.polar_closure (π•œ : Type u_1) [nontrivially_normed_field π•œ] {E : Type u_2} [seminormed_add_comm_group E] [normed_space π•œ E] (s : set E) :
theorem normed_space.smul_mem_polar {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [seminormed_add_comm_group E] [normed_space π•œ E] {s : set E} {x' : normed_space.dual π•œ E} {c : π•œ} (hc : βˆ€ (z : E), z ∈ s β†’ ‖⇑x' zβ€– ≀ β€–cβ€–) :

If x' is a dual element such that the norms β€–x' zβ€– are bounded for z ∈ s, then a small scalar multiple of x' is in polar π•œ s.

theorem normed_space.polar_ball_subset_closed_ball_div {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [seminormed_add_comm_group E] [normed_space π•œ E] {c : π•œ} (hc : 1 < β€–cβ€–) {r : ℝ} (hr : 0 < r) :
theorem normed_space.polar_closed_ball {π•œ : Type u_1} {E : Type u_2} [is_R_or_C π•œ] [normed_add_comm_group E] [normed_space π•œ E] {r : 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 (π•œ : Type u_1) [nontrivially_normed_field π•œ] {E : Type u_2} [seminormed_add_comm_group E] [normed_space π•œ 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 π•œ s are bounded by a constant.