isClosed_range_iff_antilipschitz_of_injective

English

The coercion from the equivalence range to the range of the linear map remains the same as the range restriction once a suitable inj/hclo are chosen.

Русский

Переход коe из эквивалентности в диапазон линейного отображения равен ограничению диапазона при подходящих выборках inj, hclo.

LaTeX

$$$(f.equivRange\\;hinj\\;hclo) = f.rangeRestrict$$$

Lean4

/-- An injective bounded linear operator between Banach spaces has closed range
iff it is anti-Lipschitz. -/
theorem isClosed_range_iff_antilipschitz_of_injective (f : E →L[𝕜] F) (hf : Injective f) :
    IsClosed (Set.range f) ↔ ∃ K, AntilipschitzWith K f :=
  by
  refine ⟨fun h ↦ f.antilipschitz_of_injective_of_isClosed_range hf h, fun h ↦ ?_⟩
  choose K hf' using h
  exact hf'.isClosed_range f.uniformContinuous

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