norm_mkPiAlgebra — Mathlib

English

The norm of a multilinear map is unchanged when renormalizing the scalars from 𝕜 to a subfield 𝕜'. That is, restricting scalars preserves the operator norm.

Русский

Норма многоялинейного отображения не изменяется при ограничении скаляров с 𝕜 до подполей 𝕜'.

LaTeX

$$$\|f\restriction_{\mathbb{K}'}\| = \|f\|$$$

Lean4

@[simp]
theorem norm_mkPiAlgebra [NormOneClass A] : ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1 :=
  by
  cases isEmpty_or_nonempty ι
  · simp [norm_mkPiAlgebra_of_empty]
  · refine le_antisymm norm_mkPiAlgebra_le ?_
    convert ratio_le_opNorm (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun _ => 1
    simp

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