/-- Let `S` be a subspace of a finite-dimensional complex inner product space `V`.  A linear
isometry mapping `S` into `V` can be extended to a full isometry of `V`.

TODO:  The case when `S` is a finite-dimensional subspace of an infinite-dimensional `V`.-/
noncomputable def LinearIsometry.extend (L : S →ₗᵢ[𝕜] V) : V →ₗᵢ[𝕜] V := by
  -- Build an isometry from Sᗮ to L(S)ᗮ through euclidean_space
  let d := finrank 𝕜 Sᗮ
  let LS := LinearMap.range L.toLinearMap
  have E : Sᗮ ≃ₗᵢ[𝕜] LSᗮ := by
    have dim_LS_perp : finrank 𝕜 LSᗮ = d :=
      calc
        finrank 𝕜 LSᗮ = finrank 𝕜 V - finrank 𝕜 LS := by
          simp only [← LS.finrank_add_finrank_orthogonal, add_tsub_cancel_left]
        _ = finrank 𝕜 V - finrank 𝕜 S := by simp only [LinearMap.finrank_range_of_inj L.injective]
        _ = finrank 𝕜 Sᗮ := by simp only [← S.finrank_add_finrank_orthogonal, add_tsub_cancel_left]

    exact
      (stdOrthonormalBasis 𝕜 Sᗮ).repr.trans
        ((stdOrthonormalBasis 𝕜 LSᗮ).reindex <| finCongr dim_LS_perp).repr.symm
  let L3 := LSᗮ.subtypeₗᵢ.comp E.toLinearIsometry
  -- Project onto S and Sᗮ
  haveI : CompleteSpace S := FiniteDimensional.complete 𝕜 S
  haveI : CompleteSpace V := FiniteDimensional.complete 𝕜 V
  let p1 := (orthogonalProjection S).toLinearMap
  let p2 := (orthogonalProjection Sᗮ).toLinearMap
  -- Build a linear map from the isometries on S and Sᗮ
  let M := L.toLinearMap.comp p1 + L3.toLinearMap.comp p2
  -- Prove that M is an isometry
  have M_norm_map : ∀ x : V, ‖M x‖ = ‖x‖ := by
    intro x
    -- Apply M to the orthogonal decomposition of x
    have Mx_decomp : M x = L (p1 x) + L3 (p2 x) := by
      simp only [LinearMap.add_apply, LinearMap.comp_apply, LinearMap.comp_apply,
        LinearIsometry.coe_toLinearMap]
    -- Mx_decomp is the orthogonal decomposition of M x
    have Mx_orth : ⟪L (p1 x), L3 (p2 x)⟫ = 0 := by
      have Lp1x : L (p1 x) ∈ LinearMap.range L.toLinearMap :=
        LinearMap.mem_range_self L.toLinearMap (p1 x)
      have Lp2x : L3 (p2 x) ∈ (LinearMap.range L.toLinearMap)ᗮ := by
        simp only [LinearIsometry.coe_comp, Function.comp_apply, Submodule.coe_subtypeₗᵢ, ←
          Submodule.range_subtype LSᗮ]
        apply LinearMap.mem_range_self
      

Proof state:
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝¹² : IsROrC 𝕜
E✝ : Type u_4
inst✝¹¹ : NormedAddCommGroup E✝
inst✝¹⁰ : InnerProductSpace 𝕜 E✝
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
F : Type u_6
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ F
F' : Type u_7
inst✝⁵ : NormedAddCommGroup F'
inst✝⁴ : InnerProductSpace ℝ F'
inst✝³ : Fintype ι
V : Type u_8
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
S : Submodule 𝕜 V
L✝ L : ↥S →ₗᵢ[𝕜] V
d : ℕ := finrank 𝕜 ↥Sᗮ
LS : Submodule 𝕜 V := LinearMap.range L.toLinearMap
E : ↥Sᗮ ≃ₗᵢ[𝕜] ↥LSᗮ
L3 : ↥Sᗮ →ₗᵢ[𝕜] V := comp (subtypeₗᵢ LSᗮ) (LinearIsometryEquiv.toLinearIsometry E)
this✝ : CompleteSpace ↥S
this : CompleteSpace V
p1 : V →ₗ[𝕜] ↥S := ↑(orthogonalProjection S)
p2 : V →ₗ[𝕜] ↥Sᗮ := ↑(orthogonalProjection Sᗮ)
M : V →ₗ[𝕜] V := L.toLinearMap ∘ₗ p1 + L3.toLinearMap ∘ₗ p2
x : V
Mx_decomp : M x = L (p1 x) + L3 (p2 x)
Lp1x : L (p1 x) ∈ LinearMap.range L.toLinearMap
Lp2x : L3 (p2 x) ∈ (LinearMap.range L.toLinearMap)ᗮ
⊢ inner (L (p1 x)) (L3 (p2 x)) = 0

Next tactic:
apply Submodule.inner_right_of_mem_orthogonal Lp1x Lp2x