le_closure — Mathlib

English

If f ≤ g and g is closable, then f.closure ≤ g.closure.

Русский

Если f ⊆ g и g замыкаемо, то f.closure ⊆ g.closure.

LaTeX

$$$f \le g \implies f.closure \le g.closure$ \\text{(assuming } g \\text{ is closable)}$$

Lean4

/-- A `LinearPMap` is contained in its closure. -/
theorem le_closure (f : E →ₗ.[R] F) : f ≤ f.closure :=
  by
  by_cases hf : f.IsClosable
  · refine le_of_le_graph ?_
    rw [← hf.graph_closure_eq_closure_graph]
    exact (graph f).le_topologicalClosure
  rw [closure_def' hf]

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