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[Merged by Bors] - chore(analysis/normed_space/dual): golf some proofs #8694

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[Merged by Bors] - chore(analysis/normed_space/dual): golf some proofs #8694

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42 changes: 14 additions & 28 deletions src/analysis/normed_space/dual.lean
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Expand Up @@ -52,42 +52,28 @@ variables (E : Type*) [semi_normed_group E] [semi_normed_space 𝕜 E]
variables (F : Type*) [normed_group F] [normed_space 𝕜 F]

/-- The topological dual of a seminormed space `E`. -/
@[derive [has_coe_to_fun, semi_normed_group, semi_normed_space 𝕜]] def dual := E →L[𝕜] 𝕜

instance : inhabited (dual 𝕜 E) := ⟨0⟩
@[derive [inhabited, has_coe_to_fun, semi_normed_group, semi_normed_space 𝕜]] def dual := E →L[𝕜] 𝕜

instance : normed_group (dual 𝕜 F) := continuous_linear_map.to_normed_group

instance : normed_space 𝕜 (dual 𝕜 F) := continuous_linear_map.to_normed_space

/-- The inclusion of a normed space in its double (topological) dual. -/
def inclusion_in_double_dual' (x : E) : (dual 𝕜 (dual 𝕜 E)) :=
linear_map.mk_continuous
{ to_fun := λ f, f x,
map_add' := by simp,
map_smul' := by simp }
∥x∥
(λ f, by { rw mul_comm, exact f.le_op_norm x } )

@[simp] lemma dual_def (x : E) (f : dual 𝕜 E) :
((inclusion_in_double_dual' 𝕜 E) x) f = f x := rfl

lemma double_dual_bound (x : E) : ∥(inclusion_in_double_dual' 𝕜 E) x∥ ≤ ∥x∥ :=
begin
apply continuous_linear_map.op_norm_le_bound,
{ simp },
{ intros f, rw mul_comm, exact f.le_op_norm x, }
end

/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusion_in_double_dual : E →L[𝕜] (dual 𝕜 (dual 𝕜 E)) :=
linear_map.mk_continuous
{ to_fun := λ (x : E), (inclusion_in_double_dual' 𝕜 E) x,
map_add' := λ x y, by { ext, simp },
map_smul' := λ (c : 𝕜) x, by { ext, simp } }
1
(λ x, by { convert double_dual_bound _ _ _, simp } )
continuous_linear_map.apply 𝕜 𝕜

@[simp] lemma dual_def (x : E) (f : dual 𝕜 E) : inclusion_in_double_dual 𝕜 E x f = f x := rfl

lemma inclusion_in_double_dual_norm_eq :
∥inclusion_in_double_dual 𝕜 E∥ = ∥(continuous_linear_map.id 𝕜 (dual 𝕜 E))∥ :=
continuous_linear_map.op_norm_flip _

lemma inclusion_in_double_dual_norm_le : ∥inclusion_in_double_dual 𝕜 E∥ ≤ 1 :=
by { rw inclusion_in_double_dual_norm_eq, exact continuous_linear_map.norm_id_le }

lemma double_dual_bound (x : E) : ∥(inclusion_in_double_dual 𝕜 E) x∥ ≤ ∥x∥ :=
by simpa using continuous_linear_map.le_of_op_norm_le _ (inclusion_in_double_dual_norm_le 𝕜 E) x

end general

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