The Hahn-Banach theorem applied to the Kelvin-Planck Second Law yields entropy and temperature functions satisfying the Clausius-Duhem inequality for every process, with existence independent of equilibrium but uniqueness requiring reversible access to all states.
1961 A method of geometrical representation of the thermo- dynamic properties of substances by means of surfaces
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How the Hahn-Banach Theorem Sheds Bright Light on Fundamental Questions in Classical Thermodynamics
The Hahn-Banach theorem applied to the Kelvin-Planck Second Law yields entropy and temperature functions satisfying the Clausius-Duhem inequality for every process, with existence independent of equilibrium but uniqueness requiring reversible access to all states.