In this chapter, we begin by recalling the definition of convex sets and introducing the concept of the convex hull, along with its key characterizations. We then explore the separation properties of convex sets using hyperplanes. Next, we examine convex functions and their fundamental properties. In particular, we provide equivalent characterizations of convexity for continuously differentiable functions and establish the co-coercivity result for continuously differentiable convex functions with Lipschitz continuous gradients. These results form the foundation for many developments in this book and will be used throughout.