Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures.
This point of view turned out to be particularly useful for the study of differential and integral equations.
However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra.
[1][2] The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy.
Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.
In modern introductory texts on functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite-dimensional spaces.
[3][4] In contrast, linear algebra deals mostly with finite-dimensional spaces, and does not use topology.
An important part of functional analysis is the extension of the theories of measure, integration, and probability to infinite-dimensional spaces, also known as infinite dimensional analysis.
The basic and historically first class of spaces studied in functional analysis are complete normed vector spaces over the real or complex numbers.
An important example is a Hilbert space, where the norm arises from an inner product.
These spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics, machine learning, partial differential equations, and Fourier analysis.
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.
[5] Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to
One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace.
Many special cases of this invariant subspace problem have already been proven.
In particular, many Banach spaces lack a notion analogous to an orthonormal basis.
is the counting measure, then the integral may be replaced by a sum.
Then it is not necessary to deal with equivalence classes, and the space is denoted
A general Banach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensional situation.
Also, the notion of derivative can be extended to arbitrary functions between Banach spaces.
[6] There are four major theorems which are sometimes called the four pillars of functional analysis: Important results of functional analysis include: The uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis.
The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus but it was also proven independently by Hans Hahn.
This is the beginning of the vast research area of functional analysis called operator theory; see also the spectral measure.
There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces.
The open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map.
The proof uses the Baire category theorem, and completeness of both
is a compact Hausdorff space, then the graph of a linear map
[9] Most spaces considered in functional analysis have infinite dimension.
However, a somewhat different concept, the Schauder basis, is usually more relevant in functional analysis.
