Overall, this chapter lays the groundwork and provides foundational material necessary for understanding the more complex theories and applications discussed later in the book, by providing essential tools, concepts, and results in topology, algebra, and functional analysis. There are eleven sections in this chapter, covering the following topics: §1.1 deals with topological spaces, relative topology, the concept of neighborhood, Hausdorff topological spaces, continuous functions acting between topological spaces, the product topology, the notion of local base at a point, interior and closure, compact topological spaces, relative compactness, limit points, the concept of net (and subnet), limits of nets. In §1.2 we discuss distance and metric spaces, Cauchy sequences and completeness in metric spaces. In §1.3 we introduce vector spaces, linear combinations, linear independence, the notion of basis in a vector space, the dimension of a vector space, linear transformations (operators/mappings), the concept of kernel and range for a linear operator, linear isomorphisms, algebraic complement, the concept of codimension, quotient spaces. In §1.4 we introduce normed vector spaces, the notion of p-norm and p-Banach space, semi-norms, quasi-norms, and the topology induced by a norm or a quasinorm. In §1.5 we consider inner products, Hilbert spaces, orthogonality. In §1.6 we initiate the study of linear topological spaces. The material is organized into eight subsections, addressing the following topics: local bases for vector topologies in §1.6.1, characterizations of local bases in §1.6.2, closed neighborhoods in §1.6.3, Hausdorff linear topological spaces in §1.6.4, the notion of boundedness for linear topological spaces in §1.6.5, weak boundedness in Hilbert spaces in §1.6.6, locally bounded linear topological spaces in §1.6.7, the notion of distance to a subspace of a Hilbert space in §1.6.8. Additional exercises pertaining to this material are contained in §1.7. Next, §1.8 which is concerned with finite-dimensional issues, is divided into seven subsections, presenting: linear topological completeness in §1.8.1, dissecting and reassembling linear topological spaces in §1.8.2, projections in §1.8.3, subspaces of codimension one in §1.8.4, a finite-dimensional denouement in §1.8.5, finite-dimensional extensions in §1.8.6, and a converse to the Heine-Borel theorem in §1.8.7. Once again, additional exercises concerning this material are collected in §1.9. Subsequently, the Aoki-Rolewicz theorem in topological vector spaces is discussed in §1.10, and we revisit quasi-Banach spaces in §1.11. The last section in this chapter, §1.12, is a foray into locally convex topological vector spaces. The goal here is to indicate how local convexity is an essential structural ingredient guaranteeing the existence of an abundance of continuous linear functional on the space in question.