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/- | |
Copyright (c) 2020 Heather Macbeth. All rights reserved. | |
Released under Apache 2.0 license as described in the file LICENSE. | |
Authors: Heather Macbeth | |
-/ | |
import analysis.normed_space.hahn_banach.extension | |
import analysis.normed_space.is_R_or_C | |
import analysis.locally_convex.polar | |
/-! | |
# 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 = β` or `π = 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 ββα΅’[π] (dual π (dual π 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 | |
* `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 π s` is the subset of `dual π E` consisting of those functionals `x'` for which | |
`β₯x' zβ₯ β€ 1` for every `z β s`. | |
## Tags | |
dual | |
-/ | |
noncomputable theory | |
open_locale classical topological_space | |
universes u v | |
namespace normed_space | |
section general | |
variables (π : Type*) [nontrivially_normed_field π] | |
variables (E : Type*) [seminormed_add_comm_group E] [normed_space π E] | |
variables (F : Type*) [normed_add_comm_group F] [normed_space π F] | |
/-- The topological dual of a seminormed space `E`. -/ | |
@[derive [inhabited, seminormed_add_comm_group, normed_space π]] def dual := E βL[π] π | |
instance : continuous_linear_map_class (dual π E) π E π := | |
continuous_linear_map.continuous_semilinear_map_class | |
instance : has_coe_to_fun (dual π E) (Ξ» _, E β π) := continuous_linear_map.to_fun | |
instance : normed_add_comm_group (dual π F) := continuous_linear_map.to_normed_add_comm_group | |
instance [finite_dimensional π E] : finite_dimensional π (dual π E) := | |
continuous_linear_map.finite_dimensional | |
/-- The inclusion of a normed space in its double (topological) dual, considered | |
as a bounded linear map. -/ | |
def inclusion_in_double_dual : E βL[π] (dual π (dual π E)) := | |
continuous_linear_map.apply π π | |
@[simp] lemma dual_def (x : E) (f : dual π E) : inclusion_in_double_dual π E x f = f x := rfl | |
lemma inclusion_in_double_dual_norm_eq : | |
β₯inclusion_in_double_dual π Eβ₯ = β₯(continuous_linear_map.id π (dual π E))β₯ := | |
continuous_linear_map.op_norm_flip _ | |
lemma inclusion_in_double_dual_norm_le : β₯inclusion_in_double_dual π Eβ₯ β€ 1 := | |
by { rw inclusion_in_double_dual_norm_eq, exact continuous_linear_map.norm_id_le } | |
lemma double_dual_bound (x : E) : β₯(inclusion_in_double_dual π E) xβ₯ β€ β₯xβ₯ := | |
by simpa using continuous_linear_map.le_of_op_norm_le _ (inclusion_in_double_dual_norm_le π E) x | |
/-- The dual pairing as a bilinear form. -/ | |
def dual_pairing : (dual π E) ββ[π] E ββ[π] π := continuous_linear_map.coe_lm π | |
@[simp] lemma dual_pairing_apply {v : dual π E} {x : E} : dual_pairing π E v x = v x := rfl | |
lemma dual_pairing_separating_left : (dual_pairing π E).separating_left := | |
begin | |
rw [linear_map.separating_left_iff_ker_eq_bot, linear_map.ker_eq_bot], | |
exact continuous_linear_map.coe_injective, | |
end | |
end general | |
section bidual_isometry | |
variables (π : Type v) [is_R_or_C π] | |
{E : Type u} [normed_add_comm_group E] [normed_space π E] | |
/-- If one controls the norm of every `f x`, then one controls the norm of `x`. | |
Compare `continuous_linear_map.op_norm_le_bound`. -/ | |
lemma norm_le_dual_bound (x : E) {M : β} (hMp: 0 β€ M) (hM : β (f : dual π E), β₯f xβ₯ β€ M * β₯fβ₯) : | |
β₯xβ₯ β€ M := | |
begin | |
classical, | |
by_cases h : x = 0, | |
{ simp only [h, hMp, norm_zero] }, | |
{ obtain β¨f, hfβ, hfxβ© : β f : E βL[π] π, β₯fβ₯ = 1 β§ f x = β₯xβ₯ := exists_dual_vector π x h, | |
calc β₯xβ₯ = β₯(β₯xβ₯ : π)β₯ : is_R_or_C.norm_coe_norm.symm | |
... = β₯f xβ₯ : by rw hfx | |
... β€ M * β₯fβ₯ : hM f | |
... = M : by rw [hfβ, mul_one] } | |
end | |
lemma eq_zero_of_forall_dual_eq_zero {x : E} (h : β f : dual π E, f x = (0 : π)) : x = 0 := | |
norm_le_zero_iff.mp (norm_le_dual_bound π x le_rfl (Ξ» f, by simp [h f])) | |
lemma eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 β β g : dual π E, g x = 0 := | |
β¨Ξ» hx, by simp [hx], Ξ» h, eq_zero_of_forall_dual_eq_zero π hβ© | |
/-- See also `geometric_hahn_banach_point_point`. -/ | |
lemma eq_iff_forall_dual_eq {x y : E} : | |
x = y β β g : dual π E, g x = g y := | |
begin | |
rw [β sub_eq_zero, eq_zero_iff_forall_dual_eq_zero π (x - y)], | |
simp [sub_eq_zero], | |
end | |
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/ | |
def inclusion_in_double_dual_li : E ββα΅’[π] (dual π (dual π E)) := | |
{ norm_map' := begin | |
intros x, | |
apply le_antisymm, | |
{ exact double_dual_bound π E x }, | |
rw continuous_linear_map.norm_def, | |
refine le_cInf continuous_linear_map.bounds_nonempty _, | |
rintros c β¨hc1, hc2β©, | |
exact norm_le_dual_bound π x hc1 hc2 | |
end, | |
.. inclusion_in_double_dual π E } | |
end bidual_isometry | |
section polar_sets | |
open metric set normed_space | |
/-- Given a subset `s` in a normed space `E` (over a field `π`), the polar | |
`polar π s` is the subset of `dual π E` consisting of those functionals which | |
evaluate to something of norm at most one at all points `z β s`. -/ | |
def polar (π : Type*) [nontrivially_normed_field π] | |
{E : Type*} [seminormed_add_comm_group E] [normed_space π E] : set E β set (dual π E) := | |
(dual_pairing π E).flip.polar | |
variables (π : Type*) [nontrivially_normed_field π] | |
variables {E : Type*} [seminormed_add_comm_group E] [normed_space π E] | |
lemma mem_polar_iff {x' : dual π E} (s : set E) : x' β polar π s β β z β s, β₯x' zβ₯ β€ 1 := iff.rfl | |
@[simp] lemma polar_univ : polar π (univ : set E) = {(0 : dual π E)} := | |
(dual_pairing π E).flip.polar_univ | |
(linear_map.flip_separating_right.mpr (dual_pairing_separating_left π E)) | |
lemma is_closed_polar (s : set E) : is_closed (polar π s) := | |
begin | |
dunfold normed_space.polar, | |
simp only [linear_map.polar_eq_Inter, linear_map.flip_apply], | |
refine is_closed_bInter (Ξ» z hz, _), | |
exact is_closed_Iic.preimage (continuous_linear_map.apply π π z).continuous.norm | |
end | |
@[simp] lemma polar_closure (s : set E) : polar π (closure s) = polar π s := | |
((dual_pairing π E).flip.polar_antitone subset_closure).antisymm $ | |
(dual_pairing π E).flip.polar_gc.l_le $ | |
closure_minimal ((dual_pairing π E).flip.polar_gc.le_u_l s) $ | |
by simpa [linear_map.flip_flip] | |
using (is_closed_polar _ _).preimage (inclusion_in_double_dual π E).continuous | |
variables {π} | |
/-- If `x'` is a dual element such that the norms `β₯x' zβ₯` are bounded for `z β s`, then a | |
small scalar multiple of `x'` is in `polar π s`. -/ | |
lemma smul_mem_polar {s : set E} {x' : dual π E} {c : π} | |
(hc : β z, z β s β β₯ x' z β₯ β€ β₯cβ₯) : cβ»ΒΉ β’ x' β polar π s := | |
begin | |
by_cases c_zero : c = 0, { simp only [c_zero, inv_zero, zero_smul], | |
exact (dual_pairing π E).flip.zero_mem_polar _ }, | |
have eq : β z, β₯ cβ»ΒΉ β’ (x' z) β₯ = β₯ cβ»ΒΉ β₯ * β₯ x' z β₯ := Ξ» z, norm_smul cβ»ΒΉ _, | |
have le : β z, z β s β β₯ cβ»ΒΉ β’ (x' z) β₯ β€ β₯ cβ»ΒΉ β₯ * β₯ c β₯, | |
{ intros z hzs, | |
rw eq z, | |
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _), }, | |
have cancel : β₯ cβ»ΒΉ β₯ * β₯ c β₯ = 1, | |
by simp only [c_zero, norm_eq_zero, ne.def, not_false_iff, | |
inv_mul_cancel, norm_inv], | |
rwa cancel at le, | |
end | |
lemma polar_ball_subset_closed_ball_div {c : π} (hc : 1 < β₯cβ₯) {r : β} (hr : 0 < r) : | |
polar π (ball (0 : E) r) β closed_ball (0 : dual π E) (β₯cβ₯ / r) := | |
begin | |
intros x' hx', | |
rw mem_polar_iff at hx', | |
simp only [polar, mem_set_of_eq, mem_closed_ball_zero_iff, mem_ball_zero_iff] at *, | |
have hcr : 0 < β₯cβ₯ / r, from div_pos (zero_lt_one.trans hc) hr, | |
refine continuous_linear_map.op_norm_le_of_shell hr hcr.le hc (Ξ» x hβ hβ, _), | |
calc β₯x' xβ₯ β€ 1 : hx' _ hβ | |
... β€ (β₯cβ₯ / r) * β₯xβ₯ : (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa inv_div) | |
end | |
variables (π) | |
lemma closed_ball_inv_subset_polar_closed_ball {r : β} : | |
closed_ball (0 : dual π E) rβ»ΒΉ β polar π (closed_ball (0 : E) r) := | |
Ξ» x' hx' x hx, | |
calc β₯x' xβ₯ β€ β₯x'β₯ * β₯xβ₯ : x'.le_op_norm x | |
... β€ rβ»ΒΉ * r : | |
mul_le_mul (mem_closed_ball_zero_iff.1 hx') (mem_closed_ball_zero_iff.1 hx) | |
(norm_nonneg _) (dist_nonneg.trans hx') | |
... = r / r : inv_mul_eq_div _ _ | |
... β€ 1 : div_self_le_one r | |
/-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with | |
inverse radius. -/ | |
lemma polar_closed_ball {π E : Type*} [is_R_or_C π] [normed_add_comm_group E] [normed_space π E] | |
{r : β} (hr : 0 < r) : | |
polar π (closed_ball (0 : E) r) = closed_ball (0 : dual π E) rβ»ΒΉ := | |
begin | |
refine subset.antisymm _ (closed_ball_inv_subset_polar_closed_ball _), | |
intros x' h, | |
simp only [mem_closed_ball_zero_iff], | |
refine continuous_linear_map.op_norm_le_of_ball hr (inv_nonneg.mpr hr.le) (Ξ» z hz, _), | |
simpa only [one_div] using linear_map.bound_of_ball_bound' hr 1 x'.to_linear_map h z | |
end | |
/-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms | |
of all elements of the polar `polar π s` are bounded by a constant. -/ | |
lemma bounded_polar_of_mem_nhds_zero {s : set E} (s_nhd : s β π (0 : E)) : | |
bounded (polar π s) := | |
begin | |
obtain β¨a, haβ© : β a : π, 1 < β₯aβ₯ := normed_field.exists_one_lt_norm π, | |
obtain β¨r, r_pos, r_ballβ© : β (r : β) (hr : 0 < r), ball 0 r β s := | |
metric.mem_nhds_iff.1 s_nhd, | |
exact bounded_closed_ball.mono (((dual_pairing π E).flip.polar_antitone r_ball).trans $ | |
polar_ball_subset_closed_ball_div ha r_pos) | |
end | |
end polar_sets | |
end normed_space | |