From Convexification to Schwartz Linearization: A Structural View of State Extension in Infinite Dimensions

Across mathematics, economics, and physics, a recurring structural idea appears: a space of certainty states is embedded into a larger linear or convex space in order to allow richer algebraic operations. In finite-dimensional contexts this procedure takes the form of convexification, as in von Neumann’s mixed strategies or Arrow–Debreu state-preference models. In quantum mechanics, pure states are embedded into the convex set of density operators. In this article we explain how Schwartz linear algebra extends this paradigm to infinite-dimensional functional settings. We show that, while inspired by similar structural principles, Schwartz linearization is not a probabilistic convexification, but a complete complex-linear extension of spacetime and momentum space into the tempered distribution framework. This perspective clarifies the conceptual foundations of recent developments linking Hamilton–Jacobi theory, relativistic quantum mechanics, and Maxwellian field structures.