Functional analysis is the branch of mathematics studying vector spaces with a notion of limit, such as Banach and Hilbert spaces, and the linear operators acting on them. It underlies much of modern analysis and its applications.
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- Functional analysis studies infinite-dimensional vector spaces equipped with norms or inner products, including Banach and Hilbert spaces, and the continuous linear operators between them. Central results address completeness, duality, spectral theory and the structure of bounded and compact operators.
- It provides the framework for problems in differential equations, quantum mechanics, signal processing and optimisation, where functions are treated as points in a space. Many methods in numerical analysis and convex optimisation rest on its concepts.