# Functional analysis

**Functional analysis** is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functionss. It has its historical roots in the study of transformations such as the Fourier transform and in the study of differential and integral equations. 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 Volterra.

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 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 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 (ℵ_{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 are much more complicated than Hilbert spaces. There is no clear definition of what would constitute a base, for example.

For any real number *p* ≥ 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 L^{p} spaces).

In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear functionals. As in linear algebra, 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.

Here we list some 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.

## Additional Remarks

Most spaces considered in functional analysis have infinite dimension. To show the existence of a Hamel basis for such spaces requires Zorn's lemma. Many very important theorems require the Hahn-Banach theorem which itself requires Zorn's lemma in case of an infinite dimensional space. To better understand the long-term evolution of mathematics, one should have a look at how the acceptance of the axiom of choice, being equivalent to Zorn's lemma, has changed during time.

## Literature

- Dunford and Schwartz: Linear Operators (3 books), includes visualization charts