Functional analysis studies infinite-dimensional linear worlds
Functional analysis studies vector spaces equipped with notions of limit, such as norms, inner products, and topologies, along with the linear maps that respect them. Unlike ordinary linear algebra it usually works in infinitely many dimensions, and it grew from treating operations like the Fourier transform as maps between spaces of functions.
Using functional as a noun, for a function whose input is itself a function, comes from the calculus of variations, and Hadamard’s 1910 book on that subject was the first to use it. Vito Volterra had introduced the general idea in 1887. Hadamard’s students Fréchet and Lévy carried on the nonlinear theory, while the linear side was built up by Riesz and by Stefan Banach’s circle of Polish mathematicians.
The earliest spaces studied were Banach spaces: vector spaces over the reals or complexes with a norm and no missing limits. When the norm comes from an inner product the result is a Hilbert space, the setting for quantum mechanics, partial differential equations, machine learning, and Fourier analysis. Hilbert spaces are easy to classify, with exactly one for each size of orthonormal basis, and every separable infinite-dimensional one is a copy of the space of square-summable sequences. General Banach spaces resist such tidy sorting, and many lack anything like an orthonormal basis.
Four results are often called the field’s pillars. The uniform boundedness principle, published by Banach and Hugo Steinhaus in 1927 and found independently by Hans Hahn, says that for continuous linear operators on a Banach space, being bounded at each point forces a single uniform bound. The Hahn–Banach theorem extends bounded linear functionals from a subspace to the whole space, and the open mapping theorem of Banach and Juliusz Schauder shows that a surjective continuous linear operator between Banach spaces sends open sets to open sets. Spectral theorems complete the list and lead into operator theory.
One question remains open: whether each bounded linear operator acting on a Hilbert space must leave some proper subspace invariant, though many special cases are settled. The foundations lean on choice principles, since the Baire category theorem needs a form of the axiom of choice. Later directions include Alain Connes’s noncommutative geometry and Jean Bourgain’s combinatorial study of Banach space geometry.
Source: Functional analysis
