contMDiffOn — Mathlib

English

Fix a fixed target space F3. If f: M → F1 →L F2 is ContMDiff within a neighborhood, then the map y ↦ (f(y)).precomp F3 is ContMDiff into the space of bilinear maps, i.e., the precomposition operator is ContMDiff.

Русский

Пусть f: M → F1 →L F2 гладко; тогда $y \mapsto f(y).precomp F_3$ гладко в соответствующем отношении.

LaTeX

$$$\\text{If } f: M \\to F_1 \\to_L F_2 \\text{ is ContMDiffWithinAt (or ContMDiffAround), then } y \\mapsto (f(y)).precomp F_3\\text{ is ContMDiffWithinAt of the appropriate type.}$$$

Lean4

theorem contMDiffOn (L : E →L[𝕜] F) {s} : ContMDiffOn 𝓘(𝕜, E) 𝓘(𝕜, F) n L s :=
  L.contMDiff.contMDiffOn

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