isCompact_image_coe_closedBall — Mathlib

English

The identity map on a normed space is conformal.

Русский

Идентичное отображение на нормированном пространстве конформно.

LaTeX

$$$\\mathrm{IsConformalMap}(\\mathrm{id}_{R})$$$

Lean4

/-- **Banach-Alaoglu** theorem. 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 compact in the topology of
pointwise convergence. Other versions of this theorem can be found in
`Analysis.Normed.Module.WeakDual`. -/
theorem isCompact_image_coe_closedBall [ProperSpace F] (f₀ : E →SL[σ₁₂] F) (r : ℝ) :
    IsCompact (((↑) : (E →SL[σ₁₂] F) → E → F) '' closedBall f₀ r) :=
  isCompact_image_coe_of_bounded_of_weak_closed isBounded_closedBall <| is_weak_closed_closedBall f₀ r

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