English
If e is a linear equivalence between locally convex spaces and e induces an equivalence between their continuous duals, then the closure on convex sets commutes with e similarly to the previous lemma.
Русский
Если e — линейное биективное отображение между локально выпуклыми пространствами и e индуцирует эквивалентность между их непрерывными двойственными, то замыкание на выпукых множествах коммутирует с e.
LaTeX
$$theorem image_closure_of_convex {s : Set E} (hs : Convex ℝ s) (e : E ≃ₗ[𝕜] F) (he₁ : ∀ f : StrongDual 𝕜 F, Continuous (e.dualMap f)) (he₂ : ∀ f : StrongDual 𝕜 E, Continuous (e.symm.dualMap f)) : e '' (closure s) = closure (e '' s)$$
Lean4
/-- An abbreviation for indexed families of seminorms. This is mainly to allow for dot-notation. -/
abbrev SeminormFamily :=
ι → Seminorm R E
