instBoundedContinuousMapClass — Mathlib

English

The class yields a differentiable map when applied to ContDiffMap-supported functions.

Русский

Класс обеспечивает дифференцируемость для функций, поддерживаемых ContDiffMap.

LaTeX

$$$$\\text{ContDiff}(ℝ, n) f \\Rightarrow f\\text{ плавно дифференцируемый}.$$$$

Lean4

instance (B : Type*) (E F : outParam <| Type*) [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E]
    [NormedSpace ℝ F] (n : outParam ℕ∞) (K : outParam <| Compacts E) [ContDiffMapSupportedInClass B E F n K] :
    BoundedContinuousMapClass B E F where
  map_bounded
    f := by
    have := HasCompactSupport.intro K.isCompact (map_zero_on_compl f)
    rcases (map_continuous f).bounded_above_of_compact_support this with ⟨C, hC⟩
    exact map_bounded (BoundedContinuousFunction.ofNormedAddCommGroup f (map_continuous f) C hC)

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