English
The derivative of the evaluation map on ToLp is given by the canonical isomorphism; the derivative at f is the pushforward via the continuous linear equivalence between H and PiLp.
Русский
Производная отображения оценки на ToLp задается каноническим изоморфизмом; производная в f — это отображение через непрерывно-линейное эквивалентное отображение между H и PiLp.
LaTeX
$$$$ HasStrictFDerivAt\\bigl(\\text{WithLp.ofLp}\\bigr)\\bigl( (\\text{PiLp.continuousLinearEquiv } p 𝕜 E).toContinuousLinearMap f \\bigr). $$$$
Lean4
theorem differentiableOn_piLp : DifferentiableOn 𝕜 f t ↔ ∀ i, DifferentiableOn 𝕜 (fun x => f x i) t :=
by
rw [← (PiLp.continuousLinearEquiv p 𝕜 E).comp_differentiableOn_iff, differentiableOn_pi]
rfl
