English
The coproduct construction yields a continuous map under mild hypotheses.
Русский
Конструкция coproduct образует непрерывное отображение при умеренных допущениях.
LaTeX
$$$x \mapsto f(x) \coprod g(x)\text{ непрерывно}, \text{ если } f,g\text{ непрерывны.}$$$
Lean4
protected theorem isOpen [CompleteSpace E] : IsOpen (range ((↑) : (E ≃L[𝕜] F) → E →L[𝕜] F)) :=
by
rw [isOpen_iff_mem_nhds, forall_mem_range]
refine fun e => IsOpen.mem_nhds ?_ (mem_range_self _)
let O : (E →L[𝕜] F) → E →L[𝕜] E := fun f => (e.symm : F →L[𝕜] E).comp f
have h_O : Continuous O := isBoundedBilinearMap_comp.continuous_right
convert show IsOpen (O ⁻¹' {x | IsUnit x}) from Units.isOpen.preimage h_O using 1
ext f'
constructor
· rintro ⟨e', rfl⟩
exact ⟨(e'.trans e.symm).toUnit, rfl⟩
· rintro ⟨w, hw⟩
use (unitsEquiv 𝕜 E w).trans e
ext x
simp [O, hw]
