Fitzpatrick's Theorem is a result in the field of functional analysis, particularly concerning the properties of convex functions. It provides a characterization of the subdifferential of a convex function at a point, linking it to the concept of the proximal point algorithm. This theorem is useful in optimization problems where one seeks to minimize convex functions. The theorem states that if a convex function is lower semi-continuous, then its subdifferential at any point is non-empty. This means that for any point in the domain, there exists at least one supporting hyperplane, which is crucial for understanding the behavior of convex functions in optimization contexts.