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 ๐•œ = โ„ or ๐•œ = โ„‚, 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 โ†’โ‚—แตข[๐•œ] (dual ๐•œ (dual ๐•œ E)).

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

Tags #

dual

@[instance]
def normed_space.dual.inhabited (๐•œ : Type u_1) [nondiscrete_normed_field ๐•œ] (E : Type u_2) [semi_normed_group E] [semi_normed_space ๐•œ E] :
@[instance]
def normed_space.dual.semi_normed_space (๐•œ : Type u_1) [nondiscrete_normed_field ๐•œ] (E : Type u_2) [semi_normed_group E] [semi_normed_space ๐•œ E] :
semi_normed_space ๐•œ (normed_space.dual ๐•œ E)
def normed_space.dual (๐•œ : Type u_1) [nondiscrete_normed_field ๐•œ] (E : Type u_2) [semi_normed_group E] [semi_normed_space ๐•œ E] :
Type (max u_2 u_1)

The topological dual of a seminormed space E.

Equations
@[instance]
def normed_space.dual.semi_normed_group (๐•œ : Type u_1) [nondiscrete_normed_field ๐•œ] (E : Type u_2) [semi_normed_group E] [semi_normed_space ๐•œ E] :
@[instance]
def normed_space.dual.has_coe_to_fun (๐•œ : Type u_1) [nondiscrete_normed_field ๐•œ] (E : Type u_2) [semi_normed_group E] [semi_normed_space ๐•œ E] :
has_coe_to_fun (normed_space.dual ๐•œ E) (ฮป (_x : normed_space.dual ๐•œ E), E โ†’ ๐•œ)
Equations
@[instance]
def normed_space.dual.normed_group (๐•œ : Type u_1) [nondiscrete_normed_field ๐•œ] (F : Type u_3) [normed_group F] [normed_space ๐•œ F] :
Equations
@[instance]
def normed_space.dual.normed_space (๐•œ : Type u_1) [nondiscrete_normed_field ๐•œ] (F : Type u_3) [normed_group F] [normed_space ๐•œ F] :
normed_space ๐•œ (normed_space.dual ๐•œ F)
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def normed_space.inclusion_in_double_dual (๐•œ : Type u_1) [nondiscrete_normed_field ๐•œ] (E : Type u_2) [semi_normed_group E] [semi_normed_space ๐•œ E] :
E โ†’L[๐•œ] normed_space.dual ๐•œ (normed_space.dual ๐•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 (๐•œ : Type u_1) [nondiscrete_normed_field ๐•œ] (E : Type u_2) [semi_normed_group E] [semi_normed_space ๐•œ E] (x : E) (f : normed_space.dual ๐•œ E) :
theorem normed_space.norm_le_dual_bound (๐•œ : Type v) [is_R_or_C ๐•œ] {E : Type u} [normed_group E] [normed_space ๐•œ E] (x : E) {M : โ„} (hMp : 0 โ‰ค M) (hM : โˆ€ (f : normed_space.dual ๐•œ E), โˆฅโ‡‘f xโˆฅ โ‰ค M * โˆฅfโˆฅ) :

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 (๐•œ : Type v) [is_R_or_C ๐•œ] {E : Type u} [normed_group E] [normed_space ๐•œ E] {x : E} (h : โˆ€ (f : normed_space.dual ๐•œ E), โ‡‘f x = 0) :
x = 0
theorem normed_space.eq_zero_iff_forall_dual_eq_zero (๐•œ : Type v) [is_R_or_C ๐•œ] {E : Type u} [normed_group E] [normed_space ๐•œ E] (x : E) :
x = 0 โ†” โˆ€ (g : normed_space.dual ๐•œ E), โ‡‘g x = 0
theorem normed_space.eq_iff_forall_dual_eq (๐•œ : Type v) [is_R_or_C ๐•œ] {E : Type u} [normed_group E] [normed_space ๐•œ E] {x y : E} :
x = y โ†” โˆ€ (g : normed_space.dual ๐•œ E), โ‡‘g x = โ‡‘g y
def normed_space.inclusion_in_double_dual_li (๐•œ : Type v) [is_R_or_C ๐•œ] {E : Type u} [normed_group E] [normed_space ๐•œ E] :

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

Equations