ofSeqClosedGraph — Mathlib

English

A boundedness estimate on a shell yields a global bound: there exists C such that ‖f x‖ ≤ C‖x‖ for all x ≠ 0.

Русский

Границы на оболочке вокруг нуля дают глобочное ограничение: существует C such that ‖f x‖ ≤ C‖x‖ для всех x ≠ 0.

LaTeX

$$$\exists C \; (C \ge 0) \wedge \forall x \neq 0,\; \|f x\| \le C \|x\|$$$

Lean4

/-- Upgrade a `LinearMap` to a `ContinuousLinearMap` using a variation on the
**closed graph theorem**. -/
def ofSeqClosedGraph (hg : ∀ (u : ℕ → E) (x y), Tendsto u atTop (𝓝 x) → Tendsto (g ∘ u) atTop (𝓝 y) → y = g x) :
    E →L[𝕜] F where
  toLinearMap := g
  cont := g.continuous_of_seq_closed_graph hg

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