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chore(Analysis/InnerProductSpace/Basic): fix RCLike.innerProductSpace #37970

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chore(Analysis/InnerProductSpace/Basic): fix RCLike.innerProductSpace #37970

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chore(Analysis/InnerProductSpace/Basic): fix `RCLike.innerProductSpace`
JovanGerb Apr 13, 2026
ca507eb
fix whitespace
JovanGerb Apr 13, 2026
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1 change: 0 additions & 1 deletion Mathlib/Analysis/CStarAlgebra/Module/Constructions.lean
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Expand Up @@ -362,7 +362,6 @@ noncomputable instance instCStarModuleComplex : CStarModule ℂ E where
norm_eq_sqrt_norm_inner_self {x} := by
simpa only [← inner_self_re_eq_norm] using norm_eq_sqrt_re_inner x

set_option backward.isDefEq.respectTransparency false in
-- Ensures that the two ways to obtain `CStarModule ℂᵐᵒᵖ ℂ` are definitionally equal.
example : instCStarModule (A := ℂ) = instCStarModuleComplex := by with_reducible_and_instances rfl

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15 changes: 8 additions & 7 deletions Mathlib/Analysis/InnerProductSpace/Basic.lean
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Expand Up @@ -886,11 +886,11 @@ local notation "⟪" x ", " y "⟫" => inner 𝕜 x y

/-- A field `𝕜` satisfying `RCLike` is itself a `𝕜`-inner product space. -/
instance RCLike.innerProductSpace : InnerProductSpace 𝕜 𝕜 where
inner x y := y * conj x
norm_sq_eq_re_inner x := by simp only [mul_conj, ← ofReal_pow, ofReal_re]
conj_inner_symm x y := by simp only [mul_comm, map_mul, starRingEnd_self_apply]
add_left x y z := by simp only [mul_add, map_add]
smul_left x y z := by simp only [mul_comm (conj z), mul_assoc, smul_eq_mul, map_mul]
inner x y := y * star x
norm_sq_eq_re_inner x := by rw [star_def, mul_conj, ← ofReal_pow, ofReal_re]
conj_inner_symm x y := by rw [star_def, map_mul, starRingEnd_self_apply, mul_comm]
add_left x y z := by rw [star_def, map_add, mul_add]
smul_left x y z := by rw [star_def, smul_eq_mul, map_mul, mul_left_comm]

@[simp]
theorem RCLike.inner_apply (x y : 𝕜) : ⟪x, y⟫ = y * conj x :=
Expand Down Expand Up @@ -971,10 +971,11 @@ noncomputable instance RCLike.toInnerProductSpaceReal : InnerProductSpace ℝ
norm_sq_eq_re_inner := norm_sq_eq_re_inner
conj_inner_symm x y := inner_re_symm ..
add_left x y z :=
show re (_ * _) = re (_ * _) + re (_ * _) by simp only [map_add, mul_re, conj_re, conj_im]; ring
show re (_ * _) = re (_ * _) + re (_ * _) by
simp only [star_def, map_add, mul_re, conj_re, conj_im]; ring
smul_left x y r :=
show re (_ * _) = _ * re (_ * _) by
simp only [mul_re, conj_re, conj_im, conj_trivial, smul_re, smul_im]; ring
simp only [star_def, mul_re, conj_re, conj_im, conj_trivial, smul_re, smul_im]; ring

-- The instance above does not create diamonds for concrete `𝕜`:
example : (innerProductSpace : InnerProductSpace ℝ ℝ) = RCLike.toInnerProductSpaceReal := rfl
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