English
If 𝕜' is a subsingleton, vonNBoundedness transfers via subsingleton-based argument.
Русский
Если 𝕜' является пододин, перенос ограниченности выполняется через соответствующий аргумент.
LaTeX
$$$\\text{if } 𝕜' \\text{ is a subsingleton } \\operatorname{IsVonNBounded}_{\\mathbb{k}}(s) \\text{ follows from } \\operatorname{IsVonNBounded}_{\\mathbb{k}'}(s)$$$
Lean4
protected theorem restrict_scalars [NormedField 𝕜] [NormedRing 𝕜'] [NormedAlgebra 𝕜 𝕜'] [Zero E] [TopologicalSpace E]
[SMul 𝕜 E] [MulActionWithZero 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {s : Set E} (h : IsVonNBounded 𝕜' s) :
IsVonNBounded 𝕜 s :=
match subsingleton_or_nontrivial 𝕜' with
| .inl _ =>
have : Subsingleton E := MulActionWithZero.subsingleton 𝕜' E
IsVonNBounded.of_subsingleton
| .inr _ => h.restrict_scalars_of_nontrivial _
