Functional Analysis

The branch of mathematics that studies vector spaces endowed with a topology (typically normed, Banach, or inner-product/Hilbert spaces) and the continuous linear operators between them, with emphasis on infinite-dimensional phenomena and spectral properties.

The study of vector spaces endowed with topology (norms, inner products, locally convex structures) and continuous linear maps between them, with emphasis on Banach and Hilbert spaces and the central theorems that control operators' behavior.