inverse_closed_iff — Mathlib

English

If ker f = ⊥, then f.inverse is closed iff f is closed.

Русский

Если ядро f равно нулю, то обратное отображение замкнуто тогда и только тогда, когда f замкнуто.

LaTeX

$$$f.\\ker = \\{0\\} \\Rightarrow f.inverse.IsClosed \\Leftrightarrow f.IsClosed$$$

Lean4

/-- The inverse of `f : LinearPMap` is closed if and only if `f` is closed. -/
theorem inverse_closed_iff (hf : LinearMap.ker f.toFun = ⊥) : f.inverse.IsClosed ↔ f.IsClosed :=
  by
  rw [IsClosed, inverse_graph hf]
  exact (ContinuousLinearEquiv.prodComm R E F).isClosed_image

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