isClosed_image_coe_closedBall — Mathlib

English

Given f, e with dense range and IsUniformInducing, if g satisfies g ∘ e = f, then the extension is unique.

Русский

При заданных f, e с плотным диапазоном и IsUniformInducing, если g удовлетворяет g ∘ e = f, то продолжение единственно по определению.

LaTeX

$$$g \\circ e = f \\Rightarrow \\mathrm{extend}(f,e,h\\_dense,h\\_e) = g$$$

Lean4

/-- The set of functions `f : E → F` that represent continuous linear maps `f : E →SL[σ₁₂] F`
at distance `≤ r` from `f₀ : E →SL[σ₁₂] F` is closed in the topology of pointwise convergence.
This is one of the key steps in the proof of the **Banach-Alaoglu** theorem. -/
theorem isClosed_image_coe_closedBall (f₀ : E →SL[σ₁₂] F) (r : ℝ) :
    IsClosed (((↑) : (E →SL[σ₁₂] F) → E → F) '' closedBall f₀ r) :=
  isClosed_image_coe_of_bounded_of_weak_closed isBounded_closedBall (is_weak_closed_closedBall f₀ r)

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