domDomCongrEquiv — Mathlib

English

Equality of a map into a product is equivalent to equality of coordinate maps after fst and snd projections.

Русский

Равенство отображения в произведение эквивалентно равенству его координат после проекций fst и snd.

LaTeX

$$$f = g.prod h \iff (fst ∘ f) = g ∧ (snd ∘ f) = h$$$

Lean4

/-- An equivalence of the index set defines an equivalence between the spaces of continuous
multilinear maps. In case of normed spaces, this is a linear isometric equivalence, see
`ContinuousMultilinearMap.domDomCongrₗᵢ`. -/
@[simps]
def domDomCongrEquiv {ι' : Type*} (e : ι ≃ ι') :
    ContinuousMultilinearMap R (fun _ : ι => M₂) M₃ ≃ ContinuousMultilinearMap R (fun _ : ι' => M₂) M₃
    where
  toFun := domDomCongr e
  invFun := domDomCongr e.symm
  left_inv _ := ext fun _ => by simp
  right_inv _ := ext fun _ => by simp

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