/--
The space of continuous multilinear maps on `Π(i : Fin (n+1)), Ei i` is canonically isomorphic to
the space of continuous multilinear maps on `Π(i : Fin n), Ei <| castSucc i` with values in the
space of continuous linear maps on `Ei (last n)`, by separating the last variable. We register this
isomorphism as a continuous linear equiv in `continuousMultilinearCurryRightEquiv 𝕜 Ei G`.
The algebraic version (without topology) is given in `multilinearCurryRightEquiv 𝕜 Ei G`.

The direct and inverse maps are given by `f.uncurryRight` and `f.curryRight`. Use these
unless you need the full framework of linear isometric equivs.
-/
def continuousMultilinearCurryRightEquiv :
    ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G) ≃ₗᵢ[𝕜]
      ContinuousMultilinearMap 𝕜 Ei G :=
  LinearIsometryEquiv.ofBounds
    { toFun := ContinuousMultilinearMap.uncurryRight
      map_add' := fun f₁ f₂ => by
        ext m
        rfl
      map_smul' := fun c f => by
        ext m
        rfl
      invFun := ContinuousMultilinearMap.curryRight
      left_inv := ContinuousMultilinearMap.curry_uncurryRight
      right_inv := ContinuousMultilinearMap.uncurry_curryRight } (fun f => by
        

Proof state:
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)
⊢ ‖{
          toLinearMap :=
            {
              toAddHom :=
                { toFun := uncurryRight,
                  map_add' :=
                    (_ :
                      ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)),
                        uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) },
              map_smul' :=
                (_ :
                  ∀ (c : 𝕜) (f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)),
                    AddHom.toFun
                        { toFun := uncurryRight,
                          map_add' :=
                            (_ :
                              ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)),
                                uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) }
                        (c • f) =
                      (RingHom.id 𝕜) c •
                        AddHom.toFun
                          { toFun := uncurryRight,
                            map_add' :=
                              (_ :
                                ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)),
                                  uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) }
                          f) },
          invFun := curryRight,
          left_inv :=
            (_ :
              ∀ (f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)),
                curryRight (uncurryRight f) = f),
          right_inv := (_ : ∀ (f : ContinuousMultilinearMap 𝕜 Ei G), uncurryRight (curryRight f) = f) }
        f‖ ≤
    ‖f‖

Next tactic:
simp only [uncurryRight, LinearEquiv.coe_mk]