import Mathlib.Analysis.VonNeumannAlgebra.Basic
import Mathlib.Topology.Order

namespace OAI

noncomputable section
namespace FiniteFactor
variable (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

/-- The actual ultraweak topology, generated by absolutely summable vector
functionals. No continuity of a chosen trace is built into this definition. -/
@[instance_reducible] def ultraweakTopology : TopologicalSpace (H →L[ℂ] H) :=
  ⨅ (ξ : ℕ → H) (η : ℕ → H) (_ : Summable (fun n => ‖ξ n‖ * ‖η n‖)),
    TopologicalSpace.induced (fun T : H →L[ℂ] H => ∑' n, inner ℂ (ξ n) (T (η n)))
      inferInstance

variable {H}
omit [CompleteSpace H] in
theorem vectorFunctional_ultraweak_continuous [CompleteSpace H] (ξ η : H) :
    @Continuous _ _ (ultraweakTopology H) inferInstance
      (fun T : H →L[ℂ] H => inner ℂ ξ (T η)) := by
  let x : ℕ → H := fun n => if n = 0 then ξ else 0
  let y : ℕ → H := fun n => if n = 0 then η else 0
  have hs : Summable (fun n => ‖x n‖ * ‖y n‖) := by
    exact (hasSum_ite_eq (0 : ℕ) (‖ξ‖ * ‖η‖)).summable.congr (by intro n; by_cases h : n = 0 <;> simp [x,y,h])
  have hc : @Continuous _ _ (ultraweakTopology H) inferInstance
      (fun T : H →L[ℂ] H => ∑' n, inner ℂ (x n) (T (y n))) :=
    continuous_iInf_dom (i := x) (continuous_iInf_dom (i := y)
      (continuous_iInf_dom (i := hs) continuous_induced_dom))
  have he (T : H →L[ℂ] H) :
      (fun n => inner ℂ (x n) (T (y n))) =
        (fun n => if n = 0 then inner ℂ ξ (T η) else 0) := by
    funext n
    by_cases h : n = 0 <;> simp [x,y,h]
  have hf : (fun T : H →L[ℂ] H => ∑' n, inner ℂ (x n) (T (y n))) =
      (fun T => inner ℂ ξ (T η)) := by
    funext T
    rw [he]
    exact tsum_ite_eq 0 _
  rw [hf] at hc
  exact hc

end FiniteFactor

end

end OAI