A Beginner-Friendly Note on Maximal Monotone Operators

We give a self-contained and introductory account of some basic functional analytic tools needed to understand maximal monotone operators in Hilbert spaces. We review domains of (possibly unbounded) operators, closed sets and closed operators, and provide concrete examples of bounded and unbounded operators in both finite and infinite dimensions. We then explain in detail a fundamental result of Brézis: if $A$ is a maximal monotone linear operator, then its domain is dense, $A$ is closed, and $(I+\lambda A{)}^{-1}$ is a non-expansive mapping for every $\lambda >0$. The Banach fixed point theorem (contraction mapping principle) is stated and used as a key ingredient in the analysis. The presentation is aimed at beginning graduate students and readers seeing these notions for the first time.