Lema de zorn:

Abstract

The present thesis will delve into the principles of Vector Spaces, the Axiom of Choice, and Zorn's Lemma. The main focus will be on grasping the fundamentals of a vector space by utilizing Zorn's Lemma (equivalent to the Axiom of Choice) to demonstrate that every vector space possesses a base, whether finite or infinite. We will start with an introduction to vector spaces, exploring their definitions and properties and providing examples in both finite and infinite vector space contexts. Various operations and their properties will also be discussed. Moving forward, we will delve into the prerequisites of Zorn's Lemma, offering crucial definitions accompanied by illustrative examples for better comprehension. Utilizing Zorn's Lemma as a tool, we will show the existence of bases for vector spaces, whether finite or infinite, supported by robust arguments and exemplified cases. In conclusion, we will recapitulate the key points discussed, solidifying the significance of the Axiom of Choice and Zorn's Lemma in the theory of vector spaces. By doing so, we will have established a strong foundation for understanding how theoretical principles apply in practice, enhancing our comprehension of vector spaces and their bases.