OAI/Analysis/FactorGeneration/FunctionalLimits.lean

GitHub source file

14 declarations, 0 contain sorry, 0 declared as axioms

A functional bounded by a Hilbert-valued linear map factors through one inner product. The proof uses the actual range, Hahn--Banach, and Riesz.

  1. theorem OAI.Generator.Tracial.exists_inner_factor_of_bound line 24
  2. theorem OAI.Generator.Tracial.exists_bounded_functional_ultralimit line 63
  3. theorem OAI.Generator.exists_countable_total_dual line 97
  4. def OAI.Generator.Tracial.conjugateFunctional line 131
  5. theorem OAI.Generator.Tracial.conjugateFunctional_apply line 141
  6. theorem OAI.Generator.Tracial.conjugateFunctional_norm_le line 144
  7. def OAI.Generator.Tracial.hermitianFunctional line 150
  8. theorem OAI.Generator.Tracial.hermitianFunctional_apply line 153
  9. theorem OAI.Generator.Tracial.hermitianFunctional_norm_le line 159
  10. theorem OAI.Generator.Tracial.hermitianFunctional_star line 168
  11. theorem OAI.Generator.Tracial.hermitianFunctional_selfAdjoint line 173
  12. theorem OAI.Generator.Tracial.hermitianFunctional_one line 179
  13. theorem OAI.Generator.Tracial.functional_trace_of_quadratic line 185

    Source documentation

    Equality of complex quadratic values polarizes to the full trace identity.

    theorem functional_trace_of_quadratic (φ : M.toStarSubalgebra →L[ℂ] ℂ)
    (hφ : ∀ x : M.toStarSubalgebra, φ (star x*x)=φ (x*star x))
    (x y : M.toStarSubalgebra) : φ (x*y)=φ (y*x) := by
…
  14. theorem OAI.Generator.Tracial.hermitianFunctional_trace line 199