Functional analysis

Functional analysis is a branch of mathematical analysis, focusing on the study of vector spaces that have special structures like inner products, norms, or topologies. It looks at linear functions that work on these spaces while respecting their structures. This area of math grew from studying spaces made up of functions and how transformations, like the Fourier transform, act on them.

The idea of a functional—a function whose input is another function—dates back to the calculus of variations. The term was first used in a book by Hadamard in 1910, though it was introduced earlier by the mathematician Vito Volterra in 1887. Later, mathematicians like Fréchet, Lévy, Riesz, and Stefan Banach helped develop this field.

Today, functional analysis often deals with infinite-dimensional spaces, unlike linear algebra, which usually works with spaces that have a limited number of dimensions. It also extends ideas from measure, integration, and probability into these larger, more complex spaces, creating what is known as infinite dimensional analysis.

Normed vector spaces

Functional analysis studies special types of vector spaces, like Banach spaces, which are complete normed vector spaces over the real or complex numbers. These spaces are important in areas such as quantum mechanics, machine learning, and solving equations.

The study also includes Hilbert spaces, where the norm comes from an inner product, and Fréchet spaces, which are topological vector spaces without a norm. Key concepts involve continuous linear operators between these spaces, leading to structures like C*-algebras and operator algebras.

Hilbert spaces

Hilbert spaces can be classified based on the size of their orthonormal bases. Finite-dimensional ones are well understood, while infinite-dimensional separable Hilbert spaces are especially useful in applications.

Banach spaces

Banach spaces are more complex and cannot be classified as simply as Hilbert spaces. Examples include L^p-spaces, which consist of functions whose absolute values raised to the power p have a finite integral.

The Uniform Boundedness Principle and the Hahn–Banach theorem are fundamental results in this field. The Uniform Boundedness Principle states that for a family of continuous linear operators on a Banach space, pointwise boundedness implies uniform boundedness. The Hahn–Banach theorem allows extending bounded linear functionals from a subspace to the whole space.

The Open Mapping Theorem states that a continuous linear operator between Banach spaces that is surjective is also an open map. The Closed Graph Theorem provides conditions under which a linear map with a closed graph is continuous.

Foundations of mathematics considerations

Most spaces in functional analysis are very large and have infinite dimension. To prove that these spaces have a special kind of structure called a vector space basis, mathematicians sometimes use something called Zorn's lemma. Another idea, called the Schauder basis, is often more useful in this area of study. Many important results in functional analysis depend on the Hahn–Banach theorem, which is usually shown using the axiom of choice. A slightly weaker idea called the Boolean prime ideal theorem is sometimes enough instead. Another key idea, the Baire category theorem, which helps prove many important results, also needs a form of the axiom of choice.

Points of view

Functional analysis has many different ways of looking at things. One way is called abstract analysis, which studies special types of mathematical structures like topological groups, rings, and vector spaces. Another area looks at Banach spaces and their geometry, exploring topics like how chance and probability work in these spaces.

There is also noncommutative geometry, which was developed by Alain Connes and builds on ideas from earlier mathematicians. Functional analysis also connects with quantum mechanics, either in a narrow sense within mathematical physics or more broadly to include many types of representation theory.