feat(LipschitzSmooth/Convex): convex iff equivalences for K-smoothness#31
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Add the four pairwise iff equivalences for a differentiable convex function on a Hilbert space — `LipschitzSmoothWith`, `LipschitzWith (fderiv ℝ f)`, `LipschitzWith (∇ f)`, `CocoerciveWith` — synthesising the Baillon-Haddad hard direction (`LipschitzSmoothWith K f → CocoerciveWith K f`), the non-convex Hilbert-form descent lemma, and the elementary direction `CocoerciveWith → LipschitzWith (∇ f)`.
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Folded back into #29 — the four iffs are 30 lines of pure compositions of (i) Baillon-Haddad and (ii) the descent lemma in Hilbert form, and conceptually inseparable from the BH theorem itself. Splitting them was over-application of 'minimize formal dependencies'. |
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Adds the four pairwise iff equivalences for a differentiable convex function on a Hilbert space, between
LipschitzSmoothWith K f,LipschitzWith K (fderiv ℝ f),LipschitzWith K (∇ f), andCocoerciveWith K f. The substantive work lives in the prerequisite PRs; this PR is the synthesis step that composes them.This is the synthesis cap of the
LipschitzSmoothstack: each iff direction is a one-line composition of (i) the Baillon-Haddad hard directionLipschitzSmoothWith K f → CocoerciveWith K funder convexity, (ii) the non-convex Hilbert-form descent lemmaLipschitzWith K (∇ f) → LipschitzSmoothWith K f, (iii) the elementary directionCocoerciveWith K f → LipschitzWith K (∇ f)(Cauchy-Schwarz), and (iv) the Riesz isomorphismLipschitzWith K (fderiv ℝ f) ↔ LipschitzWith K (∇ f). Splitting these out of the Baillon-Haddad PR keeps that PR focused on the actual theorem and isolates the dependency on the descent lemma to this synthesis layer.Diff for the changes just in this PR over its predecessors: link