constFormalMultilinearSeries_zero

English

The constant formal multilinear series with c does not exist unless c is zero; equivalently, it is identically zero when c = 0.

Русский

Константная формальная серия равна нулю, если c = 0; иначе она имеет нулевые higher-degree_term.

LaTeX

$$$ constFormalMultilinearSeries 𝕜 E 0 = 0 $$$

Lean4

@[simp]
theorem constFormalMultilinearSeries_zero [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F]
    [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] : constFormalMultilinearSeries 𝕜 E (0 : F) = 0 :=
  by
  ext n x
  simp only [FormalMultilinearSeries.zero_apply, ContinuousMultilinearMap.zero_apply, constFormalMultilinearSeries]
  induction n
  · simp only [ContinuousMultilinearMap.uncurry0_apply]
  · simp only [constFormalMultilinearSeries.match_1.eq_2, ContinuousMultilinearMap.zero_apply]

Похожие утверждения