Functional analysis

Functional analysis

Functional analysis is a part of mathematical analysis. It studies special kinds of spaces called vector spaces. These spaces have extra features like inner products, norms, or topologies. Functional analysis looks at linear functions that work on these spaces.

This area of math began when people studied spaces made of functions. They looked at how changes, like the Fourier transform, act on these spaces.

The idea of a functional—a function that takes another function as input—started with the calculus of variations. The word was first used in a book by Hadamard in 1910. But the idea was introduced earlier by the mathematician Vito Volterra in 1887. Later, mathematicians like Fréchet, Lévy, Riesz, and Stefan Banach helped grow this field.

Today, functional analysis often works with spaces that have endless dimensions. This is different from linear algebra, which usually looks at spaces with a limited number of dimensions. It also uses ideas from measure, integration, and probability in these bigger spaces. This creates what is called infinite dimensional analysis.

Normed vector spaces

Functional analysis studies special types of vector spaces, like Banach spaces. These are complete normed vector spaces and are important in areas such as quantum mechanics and machine learning.

The study also includes Hilbert spaces and Fréchet spaces. Key concepts involve continuous linear operators between these spaces.

Hilbert spaces

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

Banach spaces

Banach spaces are more complex. Examples include L^p-spaces, which consist of functions.

The Uniform Boundedness Principle and the Hahn–Banach theorem are important 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 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 these spaces have a special structure called a vector space basis, mathematicians use something called Zorn's lemma. Another idea, the Schauder basis, is often more useful. Many important results in functional analysis depend on the Hahn–Banach theorem, which usually uses the axiom of choice. A weaker idea called the Boolean prime ideal theorem can sometimes be used instead. The Baire category theorem helps prove many results and 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. It studies special types of mathematical structures.

Another area looks at Banach spaces. It explores how chance and probability work in these spaces.

There is also noncommutative geometry. It was developed by Alain Connes.

Functional analysis connects with quantum mechanics. This can be in a narrow sense within mathematical physics or more broadly to include many types of representation theory.