radius_compContinuousLinearMap_linearIsometryEquiv_eq

English

The radius of the sum of two multilinear series satisfies the same min bound as above.

Русский

Радиус суммы двух многочленных рядов удовлетворяет тем же минимальным неравенствам.

LaTeX

$$$\\text{radius}(p+q) \\ge \\min\\{radius(p), radius(q)\\}.$$$

Lean4

@[simp]
theorem radius_compContinuousLinearMap_linearIsometryEquiv_eq [Nontrivial E] (p : FormalMultilinearSeries 𝕜 F G)
    (u : E ≃ₗᵢ[𝕜] F) : (p.compContinuousLinearMap u.toLinearIsometry.toContinuousLinearMap).radius = p.radius :=
  by
  refine le_antisymm ?_ <| le_radius_compContinuousLinearMap _ _
  have _ : Nontrivial F := u.symm.toEquiv.nontrivial
  convert radius_compContinuousLinearMap_le p u.toContinuousLinearEquiv
  have : u.toContinuousLinearEquiv.symm.toContinuousLinearMap = u.symm.toLinearIsometry.toContinuousLinearMap := rfl
  simp [this]

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