norm_mkPiAlgebraFin — Mathlib

English

In the zero-index case, the norm of mkPiAlgebraFin is exactly the norm of 1 in A, preserving the normalization of the construction in the empty product.

Русский

В нульовом случае норма mkPiAlgebraFin точна равна норме единицы в A, сохраняющей нормировку конструкции в пустом произведении.

LaTeX

$$$\|\mathrm{mkPiAlgebraFin}_{\mathbb{K}}(0, A)\| = \|1\|$$$

Lean4

@[simp]
theorem norm_mkPiAlgebraFin [NormOneClass A] : ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ = 1 :=
  by
  cases n
  · rw [norm_mkPiAlgebraFin_zero]
    simp
  · refine le_antisymm norm_mkPiAlgebraFin_succ_le ?_
    refine le_of_eq_of_le ?_ <| ratio_le_opNorm (ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ _) A) fun _ => 1
    simp

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