Fitzpatrick's Theorem

Fitzpatrick's Theorem provides a way to understand maximal monotone operators—specific mathematical functions used in analyzing complex systems—by representing them with specialized functions called Fitzpatrick functions. The theorem states that every maximal monotone operator can be uniquely represented as a Fitzpatrick function, which is convex and lower semicontinuous. This connection helps mathematicians study properties of these operators more easily by translating them into the language of convex analysis. Essentially, Fitzpatrick's Theorem offers a powerful tool for examining and working with certain types of functions that arise in optimization, differential equations, and related fields.