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Functional analysis is that branch of mathematics and specifically of analysis which is 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. 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 spaceThis article deals with Frechet spaces in functional analysis. For Frechet spaces in general topology, see T1 space. In functional analysis, Frechet spaces are certain topological vector spaces more general than, but with some similarities to, Banach spacs and other topological vector spaceIn mathematics, a topological vector space ''X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies ares not endowed with a norm.

An important object of study in functional analysis are the continuous linear operatorsIn mathematics, a linear transformation (also called linear operator or linear map is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it "prese defined on Banach and Hilbert spaces. These lead naturally to the definition of C*-algebraC -algebras are an important area of research in functional analysis. A C -algebra can be defined concretely as a complex algebra A of linear operators on a complex Hilbert space with two additional properties: A is a topologically closed set in the norms and other operator algebraIn functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space (such as a Banach space), which is typically required to be closed in a specified operator topology. In particular, it is a set of operas.

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphismIn mathematics, an isomorphism is a kind of interesting mapping between objects. Douglas Hofstadter provides an informal definition: :The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part for every cardinalityAlternative meaning: number of pitch classes in a set. In linguistics, cardinal numbers is the name given to number words that are used for quantity one two three , as opposed to ordinal numbers, words that are used for order first second third . See How 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 (ℵ₀) 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.

General Banach spaces are more complicated. 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

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.

See also: list of functional analysis topics.