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/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import algebra.quaternion
import analysis.inner_product_space.basic
/-!
# Quaternions as a normed algebra
In this file we define the following structures on the space `β := β[β]` of quaternions:
* inner product space;
* normed ring;
* normed space over `β`.
## Notation
The following notation is available with `open_locale quaternion`:
* `β` : quaternions
## Tags
quaternion, normed ring, normed space, normed algebra
-/
localized "notation `β` := quaternion β" in quaternion
open_locale real_inner_product_space
noncomputable theory
namespace quaternion
instance : has_inner β β := β¨Ξ» a b, (a * b.conj).reβ©
lemma inner_self (a : β) : βͺa, aβ« = norm_sq a := rfl
lemma inner_def (a b : β) : βͺa, bβ« = (a * b.conj).re := rfl
instance : inner_product_space β β :=
inner_product_space.of_core
{ inner := has_inner.inner,
conj_sym := Ξ» x y, by simp [inner_def, mul_comm],
nonneg_re := Ξ» x, norm_sq_nonneg,
definite := Ξ» x, norm_sq_eq_zero.1,
add_left := Ξ» x y z, by simp only [inner_def, add_mul, add_re],
smul_left := Ξ» x y r, by simp [inner_def] }
lemma norm_sq_eq_norm_sq (a : β) : norm_sq a = β₯aβ₯ * β₯aβ₯ :=
by rw [β inner_self, real_inner_self_eq_norm_mul_norm]
instance : norm_one_class β :=
β¨by rw [norm_eq_sqrt_real_inner, inner_self, norm_sq.map_one, real.sqrt_one]β©
@[simp, norm_cast] lemma norm_coe (a : β) : β₯(a : β)β₯ = β₯aβ₯ :=
by rw [norm_eq_sqrt_real_inner, inner_self, norm_sq_coe, real.sqrt_sq_eq_abs, real.norm_eq_abs]
@[simp, norm_cast] lemma nnnorm_coe (a : β) : β₯(a : β)β₯β = β₯aβ₯β :=
subtype.ext $ norm_coe a
noncomputable instance : normed_division_ring β :=
{ dist_eq := Ξ» _ _, rfl,
norm_mul' := Ξ» a b, by { simp only [norm_eq_sqrt_real_inner, inner_self, norm_sq.map_mul],
exact real.sqrt_mul norm_sq_nonneg _ } }
noncomputable instance : normed_algebra β β :=
{ norm_smul_le := Ξ» a x, (norm_smul a x).le,
to_algebra := quaternion.algebra }
instance : has_coe β β := β¨Ξ» z, β¨z.re, z.im, 0, 0β©β©
@[simp, norm_cast] lemma coe_complex_re (z : β) : (z : β).re = z.re := rfl
@[simp, norm_cast] lemma coe_complex_im_i (z : β) : (z : β).im_i = z.im := rfl
@[simp, norm_cast] lemma coe_complex_im_j (z : β) : (z : β).im_j = 0 := rfl
@[simp, norm_cast] lemma coe_complex_im_k (z : β) : (z : β).im_k = 0 := rfl
@[simp, norm_cast] lemma coe_complex_add (z w : β) : β(z + w) = (z + w : β) := by ext; simp
@[simp, norm_cast] lemma coe_complex_mul (z w : β) : β(z * w) = (z * w : β) := by ext; simp
@[simp, norm_cast] lemma coe_complex_zero : ((0 : β) : β) = 0 := rfl
@[simp, norm_cast] lemma coe_complex_one : ((1 : β) : β) = 1 := rfl
@[simp, norm_cast] lemma coe_real_complex_mul (r : β) (z : β) : (r β’ z : β) = βr * βz :=
by ext; simp
@[simp, norm_cast] lemma coe_complex_coe (r : β) : ((r : β) : β) = r := rfl
/-- Coercion `β ββ[β] β` as an algebra homomorphism. -/
def of_complex : β ββ[β] β :=
{ to_fun := coe,
map_one' := rfl,
map_zero' := rfl,
map_add' := coe_complex_add,
map_mul' := coe_complex_mul,
commutes' := Ξ» x, rfl }
@[simp] lemma coe_of_complex : βof_complex = coe := rfl
end quaternion
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