Functional Analysis
Functional Analysis is a branch of mathematical analysis that extends the methods of linear algebra and calculus from finite-dimensional vector spaces to infinite-dimensional spaces whose elements are functions. It treats entire functions as single points and studies these function spaces (such as Banach spaces and Hilbert spaces) by equipping them with a notion of size or distance, known as a norm. This framework allows for the rigorous study of concepts like convergence, continuity, and transformations (operators) in an infinite-dimensional setting, providing the essential mathematical language for theories like quantum mechanics, partial differential equations, and signal processing.
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2. Normed Vector Spaces