Functional analysis

Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. It has its historical roots in the study of transformations, such as the Fourier transform, and in the study of differential and integral equations. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to mathematician and physicist Vito Volterra and its founding is largely attributed to mathematician Stefan Banach.

Normed vector spaces
In the modern view, functional analysis is seen as the study of complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm arises from an inner product. These spaces are of fundamental importance in the mathematical formulation of quantum mechanics. More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm.

An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces. These lead naturally to the definition of C*-algebras and other operator algebras.

Hilbert spaces
Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the base. Since finite-dimensional Hilbert spaces are fully understood in linear algebra, and since morphisms of Hilbert spaces can always be divided into morphisms of spaces with Aleph-null (&alefsym;0) dimensionality, functional analysis of Hilbert spaces mostly deals with the unique Hilbert space of dimensionality Aleph-null, and its morphisms. One of the open problems in functional analysis is to prove that every operator on a Hilbert space has a proper subspace which is invariant. Many special cases have already been proven.

Banach spaces
General Banach spaces are more complicated. There is no clear definition of what would constitute a base, for example.

For any real number p &ge; 1, an example of a Banach space is given by "all Lebesgue-measurable functions whose absolute value's p-th power has finite integral" (see Lp spaces).

In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear functionals. The dual of the dual is not always isomorphic to the original space, but there is always a natural monomorphism from a space into its dual's dual. This is explained in the dual space article.

The notion of derivative is extended to arbitrary functions between Banach spaces. It turns out that the derivative of a function at a certain point is really a continuous linear map.

Major and foundational results
These are important results of functional analysis:


 * The uniform boundedness principle is a result on sets of operators with tight bounds.
 * One spectral theorem (there are more of them) gives an integral formula for normal operators on a Hilbert space. It is of central importance in the mathematical formulation of quantum mechanics.
 * The Hahn-Banach theorem is about extending functionals from a subspace to the full space, in a norm-preserving fashion. Another implication is the non-triviality of dual spaces.
 * The open mapping theorem and closed graph theorem.

See also: List of functional analysis topics.

Foundations of mathematics considerations
Most spaces considered in functional analysis have infinite dimension. To show the existence of a vector space basis for such spaces may require Zorn's lemma. Many very important theorems require the Hahn-Banach theorem, which itself is a form of the axiom of choice that is strictly weaker than the Boolean prime ideal theorem.

Points of view
Functional analysis as it currently stands includes a number of directions:


 * soft analysis, the approach to mathematical analysis based generally on topological groups, topological rings and topological vector spaces;
 * geometry of Banach spaces, a combinatorial approach as in the work of Jean Bourgain;
 * noncommutative geometry as developed by Alain Connes, based partly on previous ideas such as George Mackey's approach to ergodic theory;
 * connection with quantum mechanics, narrowly defined in mathematical physics or broadly interpreted as by Israel Gelfand to include most types of representation theory.