Functional analysis is a branch of mathematics that studies spaces of functions and their properties. It focuses on understanding how functions behave and interact, often using concepts from linear algebra and topology. This field is essential for solving differential equations and optimization problems, making it applicable in various scientific and engineering disciplines. One of the key components of functional analysis is the study of Banach spaces and Hilbert spaces, which are types of complete normed vector spaces. These spaces provide a framework for analyzing convergence, continuity, and linear operators, helping mathematicians and scientists model complex systems effectively.