chore(analysis/normed_space/hahn_banach): remove norm' (#10527)

For a normed space over `ℝ`-algebra `𝕜`, `norm'` is currently defined to be `algebra_map ℝ 𝕜 ∥x∥`.  I believe this was introduced before the `is_R_or_C` construct (including the coercion from `ℝ` to `𝕜` for `[is_R_or_C 𝕜]`) joined mathlib.  Now that we have these things, it's easy to just say `(∥x∥ : 𝕜)` instead of `norm' 𝕜 ∥x∥`, so I don't really think `norm'` needs to exist any more.

(In principle, `norm'` is more general, since it works for all `ℝ`-algebras `𝕜` rather than just `[is_R_or_C 𝕜]`.  But I can only really think of applications in the`is_R_or_C` case, and that's the only way it's used in the library.)

src/analysis/normed_space/dual.lean

@@ -80,7 +80,7 @@ begin
   by_cases h : x = 0,
   { simp only [h, hMp, norm_zero] },
   { obtain ⟨f, hf⟩ : ∃ g : E →L[𝕜] 𝕜, _ := exists_dual_vector 𝕜 x h,
-    calc ∥x∥ = ∥norm' 𝕜 x∥ : (norm_norm' _ _ _).symm
+    calc ∥x∥ = ∥(∥x∥ : 𝕜)∥ : is_R_or_C.norm_coe_norm.symm
     ... = ∥f x∥ : by rw hf.2
     ... ≤ M * ∥f∥ : hM f
     ... = M : by rw [hf.1, mul_one] }

src/analysis/normed_space/hahn_banach.lean

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Heather Macbeth
 -/
 import analysis.convex.cone
+import analysis.normed_space.is_R_or_C
 import analysis.normed_space.extend
 
 /-!
@@ -23,32 +24,6 @@ of `𝕜`).
 
 universes u v
 
-/--
-The norm of `x` as an element of `𝕜` (a normed algebra over `ℝ`). This is needed in particular to
-state equalities of the form `g x = norm' 𝕜 x` when `g` is a linear function.
-
-For the concrete cases of `ℝ` and `ℂ`, this is just `∥x∥` and `↑∥x∥`, respectively.
--/
-noncomputable def norm' (𝕜 : Type*) [nondiscrete_normed_field 𝕜] [semi_normed_algebra ℝ 𝕜]
-  {E : Type*} [semi_normed_group E] (x : E) : 𝕜 :=
-algebra_map ℝ 𝕜 ∥x∥
-
-lemma norm'_def (𝕜 : Type*) [nondiscrete_normed_field 𝕜] [semi_normed_algebra ℝ 𝕜]
-  {E : Type*} [semi_normed_group E] (x : E) :
-  norm' 𝕜 x = (algebra_map ℝ 𝕜 ∥x∥) := rfl
-
-lemma norm_norm'
-  (𝕜 : Type*) [nondiscrete_normed_field 𝕜] [semi_normed_algebra ℝ 𝕜]
-  (A : Type*) [semi_normed_group A]
-  (x : A) : ∥norm' 𝕜 x∥ = ∥x∥ :=
-by rw [norm'_def, norm_algebra_map_eq, norm_norm]
-
-@[simp] lemma norm'_eq_zero_iff
-  (𝕜 : Type*) [nondiscrete_normed_field 𝕜] [semi_normed_algebra ℝ 𝕜]
-  (A : Type*) [normed_group A] (x : A) :
-  norm' 𝕜 x = 0 ↔ x = 0 :=
-by simp [norm', ← norm_eq_zero, norm_algebra_map_eq]
-
 namespace real
 variables {E : Type*} [semi_normed_group E] [semi_normed_space ℝ E]
 
@@ -127,41 +102,41 @@ variables {E : Type u} [normed_group E] [normed_space 𝕜 E]
 open continuous_linear_equiv submodule
 open_locale classical
 
-lemma coord_norm' (x : E) (h : x ≠ 0) : ∥norm' 𝕜 x • coord 𝕜 x h∥ = 1 :=
-by rw [norm_smul, norm_norm', coord_norm, mul_inv_cancel (mt norm_eq_zero.mp h)]
+lemma coord_norm' {x : E} (h : x ≠ 0) : ∥(∥x∥ : 𝕜) • coord 𝕜 x h∥ = 1 :=
+by rw [norm_smul, is_R_or_C.norm_coe_norm, coord_norm, mul_inv_cancel (mt norm_eq_zero.mp h)]
 
 /-- Corollary of Hahn-Banach.  Given a nonzero element `x` of a normed space, there exists an
     element of the dual space, of norm `1`, whose value on `x` is `∥x∥`. -/
-theorem exists_dual_vector (x : E) (h : x ≠ 0) : ∃ g : E →L[𝕜] 𝕜, ∥g∥ = 1 ∧ g x = norm' 𝕜 x :=
+theorem exists_dual_vector (x : E) (h : x ≠ 0) : ∃ g : E →L[𝕜] 𝕜, ∥g∥ = 1 ∧ g x = ∥x∥ :=
 begin
   let p : submodule 𝕜 E := 𝕜 ∙ x,
-  let f := norm' 𝕜 x • coord 𝕜 x h,
+  let f := (∥x∥ : 𝕜) • coord 𝕜 x h,
   obtain ⟨g, hg⟩ := exists_extension_norm_eq p f,
   refine ⟨g, _, _⟩,
   { rw [hg.2, coord_norm'] },
   { calc g x = g (⟨x, mem_span_singleton_self x⟩ : 𝕜 ∙ x) : by rw coe_mk
-    ... = (norm' 𝕜 x • coord 𝕜 x h) (⟨x, mem_span_singleton_self x⟩ : 𝕜 ∙ x) : by rw ← hg.1
-    ... = norm' 𝕜 x : by simp }
+    ... = ((∥x∥ : 𝕜) • coord 𝕜 x h) (⟨x, mem_span_singleton_self x⟩ : 𝕜 ∙ x) : by rw ← hg.1
+    ... = ∥x∥ : by simp }
 end
 
 /-- Variant of Hahn-Banach, eliminating the hypothesis that `x` be nonzero, and choosing
     the dual element arbitrarily when `x = 0`. -/
 theorem exists_dual_vector' [nontrivial E] (x : E) :
-  ∃ g : E →L[𝕜] 𝕜, ∥g∥ = 1 ∧ g x = norm' 𝕜 x :=
+  ∃ g : E →L[𝕜] 𝕜, ∥g∥ = 1 ∧ g x = ∥x∥ :=
 begin
   by_cases hx : x = 0,
   { obtain ⟨y, hy⟩ := exists_ne (0 : E),
-    obtain ⟨g, hg⟩ : ∃ g : E →L[𝕜] 𝕜, ∥g∥ = 1 ∧ g y = norm' 𝕜 y := exists_dual_vector 𝕜 y hy,
+    obtain ⟨g, hg⟩ : ∃ g : E →L[𝕜] 𝕜, ∥g∥ = 1 ∧ g y = ∥y∥ := exists_dual_vector 𝕜 y hy,
     refine ⟨g, hg.left, _⟩,
-    rw [norm'_def, hx, norm_zero, ring_hom.map_zero, continuous_linear_map.map_zero] },
+    simp [hx] },
   { exact exists_dual_vector 𝕜 x hx }
 end
 
 /-- Variant of Hahn-Banach, eliminating the hypothesis that `x` be nonzero, but only ensuring that
     the dual element has norm at most `1` (this can not be improved for the trivial
     vector space). -/
 theorem exists_dual_vector'' (x : E) :
-  ∃ g : E →L[𝕜] 𝕜, ∥g∥ ≤ 1 ∧ g x = norm' 𝕜 x :=
+  ∃ g : E →L[𝕜] 𝕜, ∥g∥ ≤ 1 ∧ g x = ∥x∥ :=
 begin
   by_cases hx : x = 0,
   { refine ⟨0, by simp, _⟩,

src/measure_theory/function/ae_eq_of_integral.lean

@@ -77,8 +77,7 @@ lemma ae_eq_zero_of_forall_dual [normed_group E] [normed_space 𝕜 E]
 begin
   let u := dense_seq E,
   have hu : dense_range u := dense_range_dense_seq _,
-  have : ∀ n, ∃ g : E →L[𝕜] 𝕜, ∥g∥ ≤ 1 ∧ g (u n) = norm' 𝕜 (u n) :=
-    λ n, exists_dual_vector'' 𝕜 (u n),
+  have : ∀ n, ∃ g : E →L[𝕜] 𝕜, ∥g∥ ≤ 1 ∧ g (u n) = ∥u n∥ := λ n, exists_dual_vector'' 𝕜 (u n),
   choose s hs using this,
   have A : ∀ (a : E), (∀ n, ⟪a, s n⟫ = (0 : 𝕜)) → a = 0,
   { assume a ha,
@@ -98,7 +97,7 @@ begin
     ... ≤ 1 * ∥u n - a∥ : continuous_linear_map.le_of_op_norm_le _ (hs n).1 _
     ... < ∥a∥ / 2 : by { rw [one_mul], rwa dist_eq_norm' at hn }
     ... < ∥u n∥ : I
-    ... = ∥s n (u n)∥ : by rw [(hs n).2, norm_norm'] },
+    ... = ∥s n (u n)∥ : by rw [(hs n).2, is_R_or_C.norm_coe_norm] },
   have hfs : ∀ n : ℕ, ∀ᵐ x ∂μ, ⟪f x, s n⟫ = (0 : 𝕜), from λ n, hf (s n),
   have hf' : ∀ᵐ x ∂μ, ∀ n : ℕ, ⟪f x, s n⟫ = (0 : 𝕜), by rwa ae_all_iff,
   exact hf'.mono (λ x hx, A (f x) hx),