Explore a groundbreaking approach to metric p-Sobolev spaces in this 42-minute lecture from the Workshop on "Synthetic Curvature Bounds for Non-Smooth Spaces: Beyond Finite Dimension" at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into a novel proof of equivalence definitions, combining smooth analysis with classical duality techniques in Convex Analysis. Discover how this strategy exemplifies a broader principle, connecting plans with barycenter, Lipschitz derivations, and normal 1-currents. Gain insights into the application of these techniques in extended metric measure spaces. Based on collaborative research with Luigi Ambrosio, Toni Ikonen, and Enrico Pasqualetto, this talk offers a deep dive into advanced mathematical concepts for those interested in metric measure spaces and related fields.