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Functional Analysis is the branch of mathematics that studies vector spaces endowed with some notion of convergence, called topological vector spaces, and mappings between them, called continuous linear operators. It focuses on the properties and structures of these spaces and operators, particularly those that are invariant under certain transformations or correspond to physical phenomena. This field has deep connections with Algebraic Geometry through the study of functional equations, especially in the context of complex analysis and algebraic curves. The tools and techniques developed in Functional Analysis have been crucial for understanding the geometric properties of these objects and their moduli spaces. Moreover, Functional Analysis provides a powerful framework for addressing problems arising in various areas of Science and Mathematics, including differential equations, dynamical systems, quantum mechanics, and harmonic analysis.