feat(Analysis/Calculus/Gradient): add toDual_gradient and companions#39202
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feat(Analysis/Calculus/Gradient): add toDual_gradient and companions#39202FordUniver wants to merge 2 commits into
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Expose the Riesz isomorphism `InnerProductSpace.toDual` directly on the gradient: four new lemmas relating `(toDual 𝕜 F) (∇ f)` to `fderiv 𝕜 f` (and the `gradientWithin` and composed versions). These are the natural identifications: the gradient is defined by applying `(toDual 𝕜 F).symm` to the Fréchet derivative, so `toDual` recovers the latter. Useful when transferring statements about `fderiv` (Lipschitz constants, etc.) to the gradient via the Riesz isometry.
PR summary 45f0cc9e9bImport changes for modified filesNo significant changes to the import graph Import changes for all files
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toDual_gradient,toDual_gradientWithin, and the composed variantstoDual_comp_gradient,toDual_comp_gradientWithin— the natural inverse direction of the gradient's defining equation∇ f x := (toDual 𝕜 F).symm (fderiv 𝕜 f x). These identify(toDual 𝕜 F) (∇ f x)withfderiv 𝕜 f x(and thegradientWithinand composed forms with the corresponding fderiv versions), making the Riesz isomorphism between the two derivative views explicit. The proofs ofDifferentiableAt.hasGradientAtandDifferentiableWithinAt.hasGradientWithinAtin the same file are simplified to use them.Came up while formalizing the descent lemma for Lipschitz-smooth functions, where being able to switch between
LipschitzWith K (fderiv ℝ f)andLipschitzWith K (∇ f)is helpful, which with this PR becomestoDual_comp_gradient ▸ (toDual ℝ F).isometry.lipschitzWith_iff K. Also slightly simplifies two call sites in mathlib.