Functional analysis

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One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Drum vibration mode12.gif
One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis.

Functional analysis is a branch of analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

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The usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian andVito 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 around Stefan Banach.

In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite-dimensional spaces. 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 theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.

Normed vector spaces

The basic and historically first class of spaces studied in functional analysis are 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 many areas, including 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 orthonormal basis. [3] Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to . Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space. 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.

Banach spaces

General Banach spaces are more complicated than Hilbert spaces, and cannot be classified in such a simple manner as those. In particular, many Banach spaces lack a notion analogous to an orthonormal basis.

Examples of Banach spaces are -spaces for any real number . Given also a measure on set , then , sometimes also denoted or , has as its vectors equivalence classes of measurable functions whose absolute value's -th power has finite integral, that is, functions for which one has

.

If is the counting measure, then the integral may be replaced by a sum. That is, we require

.

Then it is not necessary to deal with equivalence classes, and the space is denoted , written more simply in the case when is the set of non-negative integers.

In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear maps from the space into its underlying field, so-called functionals. A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space. The corresponding map is an isometry but in general not onto. A general Banach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensional situation. This is explained in the dual space article.

Also, the notion of derivative can be extended to arbitrary functions between Banach spaces. See, for instance, the Fréchet derivative article.

Major and foundational results

Important results of functional analysis include:

Uniform boundedness principle

The uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus but it was also proven independently by Hans Hahn.

Theorem (Uniform Boundedness Principle). Let X be a Banach space and Y be a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y. If for all x in X one has

then

Spectral theorem

There are many theorems known as the spectral theorem, but one in particular has many applications in functional analysis.

Theorem: [4] Let A be a bounded self-adjoint operator on a Hilbert space H. Then there is a measure space (X, Σ, μ) and a real-valued essentially bounded measurable function f on X and a unitary operator U:HL2μ(X) such that

where T is the multiplication operator:

and

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 only difference in the conclusion is that now may be complex-valued.

Hahn–Banach theorem

The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting".

Hahn–Banach theorem: [5] If p : VR is a sublinear function, and φ : UR is a linear functional on a linear subspace UV which is dominated by p on U, i.e.

then there exists a linear extension ψ : VR of φ to the whole space V, i.e., there exists a linear functional ψ such that

Open mapping theorem

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. More precisely,: [5]

Open mapping theorem. If X and Y are Banach spaces and A : XY is a surjective continuous linear operator, then A is an open map (i.e. if U is an open set in X, then A(U) is open in Y).

The proof uses the Baire category theorem, and completeness of both X and Y is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if X and Y are taken to be Fréchet spaces.

Closed graph theorem

The closed graph theorem states the following: If X is a topological space and Y is a compact Hausdorff space, then the graph of a linear map T from X to Y is closed if and only if T is continuous. [6]

Other 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. However, a somewhat different concept, Schauder basis, is usually more relevant in functional analysis. Many very important theorems require the Hahn–Banach theorem, usually proved using axiom of choice, although the strictly weaker Boolean prime ideal theorem suffices. The Baire category theorem, needed to prove many important theorems, also requires a form of axiom of choice.

Points of view

Functional analysis in its present form includes the following tendencies:

See also

Related Research Articles

In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, any vector space V has a corresponding dual vector space consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry.

Normed vector space vector space on which a norm is defined

In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that has the following properties:

  1. The zero vector, 0, has zero length; every other vector has a positive length.
  2. Multiplying a vector by a positive number changes its length without changing its direction. Moreover,
  3. The triangle inequality holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line.

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz. Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, finance, engineering, and other disciplines.

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.

In the area of mathematics known as functional analysis, a reflexive space is a Banach space that coincides with the continuous dual of its continuous dual space, both as linear space and as topological space. Reflexive Banach spaces are often characterized by their geometric properties.

In functional analysis, a discipline within mathematics, given a C*-algebra A, the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of A and certain linear functionals on A. The correspondence is shown by an explicit construction of the *-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal.

In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in, is one of the most notable results of the work of James Mercer (1883–1932). It is an important theoretical tool in the theory of integral equations; it is used in the Hilbert space theory of stochastic processes, for example the Karhunen–Loève theorem; and it is also used to characterize a symmetric positive semi-definite kernel.

In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a bounded operator, and so continuous.

In mathematics, an invariant subspace of a linear mapping T : VV from some vector space V to itself is a subspace W of V that is preserved by T; that is, T(W) ⊆ W.

In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis.

The spectrum of a linear operator that operates on a Banach space consists of all scalars such that the operator does not have a bounded inverse on . The spectrum has a standard decomposition into three parts:

In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces, typically of infinite dimension. It is one aspect of nonlinear functional analysis.

In mathematics, a nuclear space is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.

In functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.

Hilbert space inner product space that is metrically complete; a Banach space whose norm induces an inner product (follows the parallelogram identity)

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.

References

  1. acsu.buffalo.edu
  2. History of the ISBN   978-93-86279-16-3 p. 195
  3. Riesz, Frigyes; Szőkefalvi-Nagy, Béla (1990). Functional analysis (Dover ed.). New York: Dover Publications. pp. 195–199. ISBN   978-0-486-66289-3.
  4. Hall, B.C. (2013), Quantum Theory for Mathematicians, Springer, p. 147
  5. 1 2 Rudin, Walter (1991). Functional analysis . McGraw-Hill Science/Engineering/Math. ISBN   978-0-07-054236-5.
  6. Munkres, James (2000), Topology (2nd ed.), Upper Saddle River: Prentice Hall, pp. 163–172, ISBN   0-13-181629-2 , p. 171

Further reading