refactor(analysis): change the symbol for norm to align with the unicode spec (#17575)

The characters in question are:

* U+2016 DOUBLE VERTICAL LINE `‖`. The Unicode Character Database says "used in pairs to indicate norm of a matrix", for instance [here](https://www.fileformat.info/info/unicode/char/2016/index.htm) and [here](https://unicode-explorer.com/c/2016)
* U+2225 PARALLEL TO `∥`. On some platforms this renders with a forward slant!

Previously `norm` was the latter and `parallel` was the former. This change swaps them around:

* `∥x∥`, `S ‖ T`: before
* `‖x‖`, `S ∥ T`: after

Code using U+2016 `‖` for fintype cardinality has been left unchanged.

Author: eric-wieser

archive/imo/imo1972_q5.lean

@@ -16,49 +16,49 @@ Prove that `|g(x)| ≤ 1` for all `x`.
 -/
 
 /--
-This proof begins by introducing the supremum of `f`, `k ≤ 1` as well as `k' = k / ∥g y∥`. We then
+This proof begins by introducing the supremum of `f`, `k ≤ 1` as well as `k' = k / ‖g y‖`. We then
 suppose that the conclusion does not hold (`hneg`) and show that `k ≤ k'` (by
-`2 * (∥f x∥ * ∥g y∥) ≤ 2 * k` obtained from the main hypothesis `hf1`) and that `k' < k` (obtained
+`2 * (‖f x‖ * ‖g y‖) ≤ 2 * k` obtained from the main hypothesis `hf1`) and that `k' < k` (obtained
 from `hneg` directly), finally raising a contradiction with `k' < k'`.
 
 (Authored by Stanislas Polu inspired by Ruben Van de Velde).
 -/
 example (f g : ℝ → ℝ)
   (hf1 : ∀ x, ∀ y, (f(x+y) + f(x-y)) = 2 * f(x) * g(y))
-  (hf2 : ∀ y, ∥f(y)∥ ≤ 1)
+  (hf2 : ∀ y, ‖f(y)‖ ≤ 1)
   (hf3 : ∃ x, f(x) ≠ 0)
   (y : ℝ) :
-  ∥g(y)∥ ≤ 1 :=
+  ‖g(y)‖ ≤ 1 :=
 begin
-  set S := set.range (λ x, ∥f x∥),
+  set S := set.range (λ x, ‖f x‖),
   -- Introduce `k`, the supremum of `f`.
   let k : ℝ := Sup (S),
 
-  -- Show that `∥f x∥ ≤ k`.
-  have hk₁ : ∀ x, ∥f x∥ ≤ k,
+  -- Show that `‖f x‖ ≤ k`.
+  have hk₁ : ∀ x, ‖f x‖ ≤ k,
   { have h : bdd_above S, from ⟨1, set.forall_range_iff.mpr hf2⟩,
     intro x,
     exact le_cSup h (set.mem_range_self x), },
-  -- Show that `2 * (∥f x∥ * ∥g y∥) ≤ 2 * k`.
-  have hk₂ : ∀ x, 2 * (∥f x∥ * ∥g y∥) ≤ 2 * k,
+  -- Show that `2 * (‖f x‖ * ‖g y‖) ≤ 2 * k`.
+  have hk₂ : ∀ x, 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k,
   { intro x,
-    calc 2 * (∥f x∥ * ∥g y∥)
-        = ∥2 * f x * g y∥ : by simp [abs_mul, mul_assoc]
-    ... = ∥f (x + y) + f (x - y)∥ : by rw hf1
-    ... ≤ ∥f (x + y)∥ + ∥f (x - y)∥ : norm_add_le _ _
+    calc 2 * (‖f x‖ * ‖g y‖)
+        = ‖2 * f x * g y‖ : by simp [abs_mul, mul_assoc]
+    ... = ‖f (x + y) + f (x - y)‖ : by rw hf1
+    ... ≤ ‖f (x + y)‖ + ‖f (x - y)‖ : norm_add_le _ _
     ... ≤ k + k : add_le_add (hk₁ _) (hk₁ _)
     ... = 2 * k : (two_mul _).symm, },
 
   -- Suppose the conclusion does not hold.
   by_contra' hneg,
-  set k' := k / ∥g y∥,
+  set k' := k / ‖g y‖,
 
   -- Demonstrate that `k' < k` using `hneg`.
   have H₁ : k' < k,
   { have h₁ : 0 < k,
     { obtain ⟨x, hx⟩ := hf3,
       calc 0
-          < ∥f x∥ : norm_pos_iff.mpr hx
+          < ‖f x‖ : norm_pos_iff.mpr hx
       ... ≤ k : hk₁ x },
     rw div_lt_iff,
     apply lt_mul_of_one_lt_right h₁ hneg,
@@ -67,8 +67,8 @@ begin
   -- Demonstrate that `k ≤ k'` using `hk₂`.
   have H₂ : k ≤ k',
   { have h₁ : ∃ x : ℝ, x ∈ S,
-    { use ∥f 0∥, exact set.mem_range_self 0, },
-    have h₂ : ∀ x, ∥f x∥ ≤ k',
+    { use ‖f 0‖, exact set.mem_range_self 0, },
+    have h₂ : ∀ x, ‖f x‖ ≤ k',
     { intros x,
       rw le_div_iff,
       { apply (mul_le_mul_left zero_lt_two).mp (hk₂ x) },
@@ -95,28 +95,28 @@ This is a more concise version of the proof proposed by Ruben Van de Velde.
 -/
 example (f g : ℝ → ℝ)
   (hf1 : ∀ x, ∀ y, (f (x+y) + f(x-y)) = 2 * f(x) * g(y))
-  (hf2 : bdd_above (set.range (λ x, ∥f x∥)))
+  (hf2 : bdd_above (set.range (λ x, ‖f x‖)))
   (hf3 : ∃ x, f(x) ≠ 0)
   (y : ℝ) :
-  ∥g(y)∥ ≤ 1 :=
+  ‖g(y)‖ ≤ 1 :=
 begin
   obtain ⟨x, hx⟩ := hf3,
-  set k := ⨆ x, ∥f x∥,
-  have h : ∀ x, ∥f x∥ ≤ k := le_csupr hf2,
+  set k := ⨆ x, ‖f x‖,
+  have h : ∀ x, ‖f x‖ ≤ k := le_csupr hf2,
   by_contra' H,
-  have hgy : 0 < ∥g y∥,
+  have hgy : 0 < ‖g y‖,
     by linarith,
   have k_pos : 0 < k := lt_of_lt_of_le (norm_pos_iff.mpr hx) (h x),
-  have : k / ∥g y∥ < k := (div_lt_iff hgy).mpr (lt_mul_of_one_lt_right k_pos H),
-  have : k ≤ k / ∥g y∥,
-  { suffices : ∀ x, ∥f x∥ ≤ k / ∥g y∥, from csupr_le this,
+  have : k / ‖g y‖ < k := (div_lt_iff hgy).mpr (lt_mul_of_one_lt_right k_pos H),
+  have : k ≤ k / ‖g y‖,
+  { suffices : ∀ x, ‖f x‖ ≤ k / ‖g y‖, from csupr_le this,
     intro x,
-    suffices : 2 * (∥f x∥ * ∥g y∥) ≤ 2 * k,
+    suffices : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k,
       by { rwa [le_div_iff hgy, ←mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)] },
-    calc 2 * (∥f x∥ * ∥g y∥)
-        = ∥2 * f x * g y∥           : by simp [abs_mul, mul_assoc]
-    ... = ∥f (x + y) + f (x - y)∥   : by rw hf1
-    ... ≤ ∥f (x + y)∥ + ∥f (x - y)∥ : abs_add _ _
+    calc 2 * (‖f x‖ * ‖g y‖)
+        = ‖2 * f x * g y‖           : by simp [abs_mul, mul_assoc]
+    ... = ‖f (x + y) + f (x - y)‖   : by rw hf1
+    ... ≤ ‖f (x + y)‖ + ‖f (x - y)‖ : abs_add _ _
     ... ≤ 2 * k                     : by linarith [h (x+y), h (x -y)] },
   linarith,
 end

counterexamples/phillips.lean

@@ -375,7 +375,7 @@ section
 -/
 
 lemma norm_indicator_le_one (s : set α) (x : α) :
-  ∥(indicator s (1 : α → ℝ)) x∥ ≤ 1 :=
+  ‖(indicator s (1 : α → ℝ)) x‖ ≤ 1 :=
 by { simp only [indicator, pi.one_apply], split_ifs; norm_num }
 
 /-- A functional in the dual space of bounded functions gives rise to a bounded additive measure,
@@ -392,8 +392,8 @@ def _root_.continuous_linear_map.to_bounded_additive_measure
         by { ext x, simp [indicator_union_of_disjoint hst], },
       rw [this, f.map_add],
     end,
-  exists_bound := ⟨∥f∥, λ s, begin
-    have I : ∥of_normed_add_comm_group_discrete (indicator s 1) 1 (norm_indicator_le_one s)∥ ≤ 1,
+  exists_bound := ⟨‖f‖, λ s, begin
+    have I : ‖of_normed_add_comm_group_discrete (indicator s 1) 1 (norm_indicator_le_one s)‖ ≤ 1,
       by apply norm_of_normed_add_comm_group_le _ zero_le_one,
     apply le_trans (f.le_op_norm _),
     simpa using mul_le_mul_of_nonneg_left I (norm_nonneg f),
@@ -492,7 +492,7 @@ lemma countable_spf_mem (Hcont : #ℝ = aleph 1) (y : ℝ) : {x | y ∈ spf Hcon
 
 We construct a function `f` from `[0,1]` to a complete Banach space `B`, which is weakly measurable
 (i.e., for any continuous linear form `φ` on `B` the function `φ ∘ f` is measurable), bounded in
-norm (i.e., for all `x`, one has `∥f x∥ ≤ 1`), and still `f` has no Pettis integral.
+norm (i.e., for all `x`, one has `‖f x‖ ≤ 1`), and still `f` has no Pettis integral.
 
 This construction, due to Phillips, requires the continuum hypothesis. We will take for `B` the
 space of all bounded functions on `ℝ`, with the supremum norm (no measure here, we are really
@@ -582,7 +582,7 @@ begin
 end
 
 /-- The function `f Hcont : ℝ → (discrete_copy ℝ →ᵇ ℝ)` is uniformly bounded by `1` in norm. -/
-lemma norm_bound (Hcont : #ℝ = aleph 1) (x : ℝ) : ∥f Hcont x∥ ≤ 1 :=
+lemma norm_bound (Hcont : #ℝ = aleph 1) (x : ℝ) : ‖f Hcont x‖ ≤ 1 :=
 norm_of_normed_add_comm_group_le _ zero_le_one _
 
 /-- The function `f Hcont : ℝ → (discrete_copy ℝ →ᵇ ℝ)` has no Pettis integral. -/

src/analysis/ODE/gronwall.lean

@@ -9,8 +9,8 @@ import analysis.special_functions.exp_deriv
 # Grönwall's inequality
 
 The main technical result of this file is the Grönwall-like inequality
-`norm_le_gronwall_bound_of_norm_deriv_right_le`. It states that if `f : ℝ → E` satisfies `∥f a∥ ≤ δ`
-and `∀ x ∈ [a, b), ∥f' x∥ ≤ K * ∥f x∥ + ε`, then for all `x ∈ [a, b]` we have `∥f x∥ ≤ δ * exp (K *
+`norm_le_gronwall_bound_of_norm_deriv_right_le`. It states that if `f : ℝ → E` satisfies `‖f a‖ ≤ δ`
+and `∀ x ∈ [a, b), ‖f' x‖ ≤ K * ‖f x‖ + ε`, then for all `x ∈ [a, b]` we have `‖f x‖ ≤ δ * exp (K *
 x) + (ε / K) * (exp (K * x) - 1)`.
 
 Then we use this inequality to prove some estimates on the possible rate of growth of the distance
@@ -22,7 +22,7 @@ Sec. 4.5][HubbardWest-ode], where `norm_le_gronwall_bound_of_norm_deriv_right_le
 
 ## TODO
 
-- Once we have FTC, prove an inequality for a function satisfying `∥f' x∥ ≤ K x * ∥f x∥ + ε`,
+- Once we have FTC, prove an inequality for a function satisfying `‖f' x‖ ≤ K x * ‖f x‖ + ε`,
   or more generally `liminf_{y→x+0} (f y - f x)/(y - x) ≤ K x * f x + ε` with any sign
   of `K x` and `f x`.
 -/
@@ -101,7 +101,7 @@ the inequalities `f a ≤ δ` and
 `∀ x ∈ [a, b), liminf_{z→x+0} (f z - f x)/(z - x) ≤ K * (f x) + ε`, then `f x`
 is bounded by `gronwall_bound δ K ε (x - a)` on `[a, b]`.
 
-See also `norm_le_gronwall_bound_of_norm_deriv_right_le` for a version bounding `∥f x∥`,
+See also `norm_le_gronwall_bound_of_norm_deriv_right_le` for a version bounding `‖f x‖`,
 `f : ℝ → E`. -/
 theorem le_gronwall_bound_of_liminf_deriv_right_le {f f' : ℝ → ℝ} {δ K ε : ℝ} {a b : ℝ}
   (hf : continuous_on f (Icc a b))
@@ -128,13 +128,13 @@ begin
 end
 
 /-- A Grönwall-like inequality: if `f : ℝ → E` is continuous on `[a, b]`, has right derivative
-`f' x` at every point `x ∈ [a, b)`, and satisfies the inequalities `∥f a∥ ≤ δ`,
-`∀ x ∈ [a, b), ∥f' x∥ ≤ K * ∥f x∥ + ε`, then `∥f x∥` is bounded by `gronwall_bound δ K ε (x - a)`
+`f' x` at every point `x ∈ [a, b)`, and satisfies the inequalities `‖f a‖ ≤ δ`,
+`∀ x ∈ [a, b), ‖f' x‖ ≤ K * ‖f x‖ + ε`, then `‖f x‖` is bounded by `gronwall_bound δ K ε (x - a)`
 on `[a, b]`. -/
 theorem norm_le_gronwall_bound_of_norm_deriv_right_le {f f' : ℝ → E} {δ K ε : ℝ} {a b : ℝ}
   (hf : continuous_on f (Icc a b)) (hf' : ∀ x ∈ Ico a b, has_deriv_within_at f (f' x) (Ici x) x)
-  (ha : ∥f a∥ ≤ δ) (bound : ∀ x ∈ Ico a b, ∥f' x∥ ≤ K * ∥f x∥ + ε) :
-  ∀ x ∈ Icc a b, ∥f x∥ ≤ gronwall_bound δ K ε (x - a) :=
+  (ha : ‖f a‖ ≤ δ) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖ + ε) :
+  ∀ x ∈ Icc a b, ‖f x‖ ≤ gronwall_bound δ K ε (x - a) :=
 le_gronwall_bound_of_liminf_deriv_right_le (continuous_norm.comp_continuous_on hf)
   (λ x hx r hr, (hf' x hx).liminf_right_slope_norm_le hr) ha bound
 

src/analysis/ODE/picard_lindelof.lean

@@ -51,7 +51,7 @@ structure is_picard_lindelof
 (hR : 0 ≤ R)
 (lipschitz : ∀ t ∈ Icc t_min t_max, lipschitz_on_with L (v t) (closed_ball x₀ R))
 (cont : ∀ x ∈ closed_ball x₀ R, continuous_on (λ (t : ℝ), v t x) (Icc t_min t_max))
-(norm_le : ∀ (t ∈ Icc t_min t_max) (x ∈ closed_ball x₀ R), ∥v t x∥ ≤ C)
+(norm_le : ∀ (t ∈ Icc t_min t_max) (x ∈ closed_ball x₀ R), ‖v t x‖ ≤ C)
 (C_mul_le_R : (C : ℝ) * linear_order.max (t_max - t₀) (t₀ - t_min) ≤ R)
 
 /-- This structure holds arguments of the Picard-Lipschitz (Cauchy-Lipschitz) theorem. It is part of
@@ -100,7 +100,7 @@ have continuous_on (uncurry (flip v)) (closed_ball v.x₀ v.R ×ˢ Icc v.t_min v
 this.comp continuous_swap.continuous_on (preimage_swap_prod _ _).symm.subset
 
 lemma norm_le {t : ℝ} (ht : t ∈ Icc v.t_min v.t_max) {x : E} (hx : x ∈ closed_ball v.x₀ v.R) :
-  ∥v t x∥ ≤ v.C :=
+  ‖v t x‖ ≤ v.C :=
 v.is_pl.norm_le _ ht _ hx
 
 /-- The maximum of distances from `t₀` to the endpoints of `[t_min, t_max]`. -/
@@ -192,7 +192,7 @@ begin
   exact ⟨(v.proj x).2, f.mem_closed_ball _⟩
 end
 
-lemma norm_v_comp_le (t : ℝ) : ∥f.v_comp t∥ ≤ v.C :=
+lemma norm_v_comp_le (t : ℝ) : ‖f.v_comp t‖ ≤ v.C :=
 v.norm_le (v.proj t).2 $ f.mem_closed_ball _
 
 lemma dist_apply_le_dist (f₁ f₂ : fun_space v) (t : Icc v.t_min v.t_max) :
@@ -262,7 +262,7 @@ begin
   simp only [dist_eq_norm, next_apply, add_sub_add_left_eq_sub,
     ← interval_integral.integral_sub (interval_integrable_v_comp _ _ _)
       (interval_integrable_v_comp _ _ _), norm_integral_eq_norm_integral_Ioc] at *,
-  calc ∥∫ τ in Ι (v.t₀ : ℝ) t, f₁.v_comp τ - f₂.v_comp τ∥
+  calc ‖∫ τ in Ι (v.t₀ : ℝ) t, f₁.v_comp τ - f₂.v_comp τ‖
       ≤ ∫ τ in Ι (v.t₀ : ℝ) t, v.L * ((v.L * |τ - v.t₀|) ^ n / n! * d) :
     begin
       refine norm_integral_le_of_norm_le (continuous.integrable_on_interval_oc _) _,
@@ -338,7 +338,7 @@ end picard_lindelof
 
 lemma is_picard_lindelof.norm_le₀ {E : Type*} [normed_add_comm_group E]
   {v : ℝ → E → E} {t_min t₀ t_max : ℝ} {x₀ : E} {C R : ℝ} {L : ℝ≥0}
-  (hpl : is_picard_lindelof v t_min t₀ t_max x₀ L R C) : ∥v t₀ x₀∥ ≤ C :=
+  (hpl : is_picard_lindelof v t_min t₀ t_max x₀ L R C) : ‖v t₀ x₀‖ ≤ C :=
 hpl.norm_le t₀ hpl.ht₀ x₀ $ mem_closed_ball_self hpl.hR
 
 /-- Picard-Lindelöf (Cauchy-Lipschitz) theorem. -/