This article aims to offer a unifying approach to the basic theory of division algebras by presenting the research of the German-American mathematician Max August Zorn, who classified alternative division algebras. In section 1 the basic theory of real division algebras is developed. Section 2 presents the Cayley-Dickson Process, which consists of constructing an extension algebra from an algebra provided with a conjugation, similar to the construction of complex numbers from real numbers. In Section 3 presents the classical division algebras R (real), C (complex), H (quaternions) and O (octonions) and mentions some of their applications. In section 4 the main theorem is presented, which establishes that the only (except isomorphism) alternative division algebras are: R, C, H and O (Zorn’s theorem). The classification theorems of associative division algebras (Frobenius) and normed ...

The objective of this article is to categorically focus on some aspects of Functional Analysis, aiming to present a high-level methodology for research and teaching. In section 1 the notions of category and subcategory are given; The Identification Convention and the Principle of Duality are stated; Some categories are presented and important properties of their objects and morphisms are stated. In section 2, functors and natural transformations are studied, showing their importance through examples that connect algebra and topology. In section 3 the important notions of limit of a functor and of adjunction of functors are given, and some examples are considered.