In this chapter, we introduce some fundamental tools essential for our subsequent abstract approach. We begin by introducing the Weierstrass–Tonelli theorem, which provides a foundation for the variational treatment of nonlinear problems, as well as the Ekeland variational principle. Then we proceed to the theory of monotone operators, supplementing it with numerous comments and examples. We also present the Browder–Minty theorem and the surjectivity theorem for pseudomonotone, bounded, and continuous operators. We place particular emphasis on the solvability of Galerkin-type approximations, which is why the finite-dimensional case is thoroughly covered. We present only those proofs whose arguments are crucial for the later parts of this text and are vital for understanding the concept of generalized solutions and the issues related to their existence. Some examples of standard applications are also given throughout the chapter.