Functional Analysis

This group gathers theorems about infinite-dimensional vector spaces of functions, the operators that act on them, and the norms and inner products that measure them, the modern generalization of linear algebra and calculus to function spaces. It covers the foundational Banach and Hilbert space results, including the Hahn-Banach, open mapping, and closed graph theorems, fixed-point and compactness results such as the Banach fixed-point and Arzela-Ascoli theorems, and operator-theoretic results, including the Fredholm alternative, that describe when linear equations in infinite dimensions can be solved.