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Functional Analysis studies spaces of functions and their transformations, with an emphasis on properties that remain invariant under various operations, such as continuity or differentiability. It explores abstract structures like Banach spaces, Hilbert spaces, and topological vector spaces, often employing sophisticated tools from other mathematical areas like measure theory, differential equations, and group theory. By analyzing functions as elements of these spaces, this field offers a unified approach to seemingly disparate topics in mathematics and science, making it an essential component of Concrete Mathematics, and thus, the Science and Mathematics hierarchies.