dual_norm_topology_le_weak_dual_topology

English

The weak dual map is injective, hence it preserves equality: x' = y' iff toWeakDual x' = toWeakDual y'.

Русский

Слабое двойственное отображение инъективно, следовательно, x' = y' тогда и только тогда, когда toWeakDual x' = toWeakDual y'.

LaTeX

$$$toWeakDual\\;: StrongDual 𝕜 E \\to WeakDual 𝕜 E\\;\\text{ is injective}$$$

Lean4

/-- The weak-star topology is coarser than the dual-norm topology. -/
theorem dual_norm_topology_le_weak_dual_topology :
    (UniformSpace.toTopologicalSpace : TopologicalSpace (StrongDual 𝕜 E)) ≤
      (instTopologicalSpaceWeakDual .. : TopologicalSpace (WeakDual 𝕜 E)) :=
  by
  convert (@toWeakDual_continuous _ _ _ _ (by assumption)).le_induced
  exact induced_id.symm

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