Actualidad y aplicaciones de la topología y el análisis funcional

Resum

Our aim in this article is to consider Topology and Functional Analysis as closely related branches of mathematics, highlighting the role of the use of algebraic topology in the treatment of certain issues in different fields of Science, as well as the great help of using functional analysis in solving important problems that arise in these fields. With regard to Topology, in addition to a synthesis on the essential of its evolution and content, is carried out as derived from it, a properly formalized exposition of the Catastrophe Theory and the Mathematics of Chaos. Insofar as the Functional Analysis, inseparable from the Topology, we studied the topological vector spaces, the normed spaces, and among them the Banach one, the dual topological space, the operational calculation, the notion of spectrum of the operator, and the Hilbert space. Finally, the application of Functional Analysis to two specific branches of Science is addressed: Quantum Mechanics and Economics.