norm_mkPiAlgebraFin_le_of_pos — Mathlib

English

The norm of the Pi-ring construction mkPiRing equals the norm of the element z mapped by the ring, i.e., the norm of mkPiRing 𝕜 ι z is the norm of z. This consolidates the behavior of mkPiRing with the target G-norm being consistent with the input.

Русский

Норма mkPiRing 𝕜 ι z равна норме z, т.е. норма mkPiRing совпадает с нормой z.

LaTeX

$$$\|\mathrm{mkPiRing}_{\mathbb{K}}(\iota, z)\| = \|z\|$$$

Lean4

theorem norm_mkPiAlgebraFin_le_of_pos (hn : 0 < n) : ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ ≤ 1 :=
  by
  obtain ⟨n, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn.ne'
  exact norm_mkPiAlgebraFin_succ_le

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