Maximal Monotonicity via Convex Analysis

In his ‘23’ “Mathematische Probleme” lecture to the Paris International Congress in 1900, David Hilbert wrote “Besides it is an error to believe that rigor in the proof is the enemy of simplicity.” In this spirit, we use simple convex analytic methods, relying on an ingenious function due to Simon Fitzpatrick, to provide a concise proof of the maximality of the sum of two maximal monotone operators on reflexive Banach space under standard transversality conditions. Many other extension, surjectivity, convexity and local boundedness results are likewise established.