Functional analysis is a branch of mathematics formed in the 1930sTheoretical physicsIt is developed from the research of.It comprehensively uses the viewpoints of function theory, geometry and modern mathematics to study the functional, operator andLimit theory。It can be regarded as the analytic geometry andmathematical analysis 。Functional analysis in mathematical physics equations, probability theory,Computational mathematicsIt is also a mathematical tool for studying physical systems with infinite degrees of freedom.
functional analysis(Functional Analysis) is a branch of modern mathematics, belonging toAnalyticsIts main research object is functionspace。Functional analysis is based on the transformation of functions (such asFourier transformation ) and the study of differential equations andintegral equation It is developed from the research of.usefunctionalAs an expression derived fromVariational method, represents the function acting on the function.Banach(Stefan Banach) is one of the main founders of functional analysis theory, while mathematician and physicistVito Volterra (Vito Volterra) has made important contributions to the wide application of functional analysis.[1]
Axiom of choice
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Most of the spaces studied by functional analysis are infinite dimensional.In order to prove the existence of a set of bases in infinite dimensional vector space, we must useZorn lemma（Zorn's Lemma）。In addition, most of the important theorems in functional analysis are based on the Han Banach theorem, which itself is the Axiom of ChoiceWeaker than Brunson's ideal theorem(Boolean prime ideal theory).[1]
history
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background
Since the 19th century, the development of mathematics has entered a new stage.This is becauseEuclid 5th postulateThe research ofNon Euclidean geometryThis new subject;aboutalgebraic equationGeneral thinking of solution, finally established and developedgroup theory；The research on mathematical analysis has establishedset theory。These new theories have prepared the conditions for the generalization of the basic concepts and methods of classical analysis with a unified view.At this time, the concept of function is given a more general meaning. The concept of function in classical analysis refers to a corresponding relationship established between two sets of numbers.The development of modern mathematics requires the establishment of a certain correspondence between two arbitrary sets.
Due to the formation of many new departments in analytics, it is revealed that many concepts and methods of analysis, algebra and set often have similarities.For example, the roots of algebraic equations and the solutions of differential equations can be appliedSuccessive approximationAnd the existence and uniqueness conditions of the solution are also very similar.This similarity is more prominent in the integral equation theory.The emergence of functional analysis is precisely related to this situation. At first glance, there are some very unrelated things that are similar.Therefore, it inspires people to explore the general and true essence from these similar things.
The establishment of non Euclidean geometry broadened people's understanding of space,N-dimensional spaceThe emergence of geometry allows us to interpret multivariate functions as images of multidimensional space in the language of geometry.In this way, the similarity between analysis and geometry is shown, and there is a possibility to geometrize analysis.This possibility requires further generalization of geometric concepts, and finallyeuclidean space Expand to infinite dimension space.
At the beginning of the 20th century,SwedenMathematician Fredholm and French mathematicianhadamard In the published works, the seeds of generalizing analytics appeared.Then,HilbertHe Hailingzhe created“Hilbert space”Research.By the 1920s, general analysis, that is, the basic concept of functional analysis, had gradually formed in the mathematical world.ResearchInfinite dimensional linear spaceOnUniversal functionAnd operator theory, has produced a new analytical mathematics, called functional analysis.In the 1930s, functional analysis has become an independent subject in mathematics.[2]
research status
Functional analysis currently includes the following branches:
Soft analysis, which aims to use mathematical analysistopologyGroup, Topological Ring andTopological vector spaceLanguage expression.
Banach spaceThe geometric structure of is represented by a series of works of Jean Bourgain.
Noncommutative geometryThe main contributors in this direction include Alain Connes, whose part of work is based on the results of George Mackey's ergodic theory.
The theory related to quantum mechanics is calledmathematical physics From a broader perspective, as described by Israel Gelfand, it includesExpressionismMost types of questions.[2]
Topological linear space
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Since functional analysis originates from studying various function spaces, there are different types of convergence of function sequences in function spaces (such as point by point convergence, uniform convergence, weak convergence, etc.), which indicates that there are different topologies in function spaces.The function space is generally an infinite dimensional linear space.Therefore, abstract functional analysis studies general (infinite dimensional) linear spaces with a certain topology.
Topological linear spaceThe definition of is a linear space with topological structure, so that the addition and number multiplication of linear space are continuous mapping spaces.[1]
Banach space
This is the most common and widely used type of topological linear space.For example, continuous function space on finite closed interval, k-degree differentiable function space on finite closed interval.Or for eachreal numberp，If p ≥ 1, oneBanach spaceThe example oflebesgue Measurable function ".
In Banach space, a considerable part of the research involves the concept of dual space, that is, the space formed by all continuous linear functionals in Banach space.The dual space of the dual space may not be isomorphic with the original space, but it can always construct a simplehomomorphism。
differentialThe concept of can be generalized in Banach space. The differential operator acts on all functions on it. The differential of a function at a given point is continuousLinear mapping。[1]
Hilbert space
Hilbert spaceThe following conclusion can be used for complete classification, that is, for any two Hilbert spaces, if their bases have the same cardinality, they must be isomorphic to each other.For a finite dimensional Hilbert space, the continuity on itlinearOperator islinear algebraStudied inlinear transformation 。For an infinite dimensional Hilbert space, anyMorphismsAll can be decomposed into morphisms on countable dimensions (base is 50), so functional analysis mainly studies Hilbert spaces and their morphisms on countable dimensions.One of the unsolved problems in Hilbert space is whether there is a truth for every operator in Hilbert spaceInvariant subspace。The answer to this question is yes under certain circumstances.[1]
operator
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In the specific function space, we have various operations on functions.The most typical operation is the derivative of a function.Such operations are generally called operators.As a mapping between topological spaces, we can always require operators to be continuous.The study of operators on topological linear spaces constitutes a major branch of functional analysis.[1]
Linear Operators and Linear Functional
The most basic operators are those that maintain the topological linear space structure, called linear operators.If a space is the number field of a topological linear space (in particular, a one-dimensional topological linear space), then such an operator becomes a linear functional.
There are several very basic and important theorems in the theory of linear operators.
1. Uniform boundedness theorem (also calledresonance theorem ）This theorem describes the properties of a family of bounded operators.
2. Rare-Banach theorem(Hahn Banach Theorem) studied how to make an operator norm preserving from a subspaceContinuationTo the whole space.Another related result isDual spaceThe extraordinary nature of.
fourSpectral theoremIt includes a series of results, of which the most commonly used result gives an integral expression of normal operators on Hilbert space, which plays a central role in the mathematical description of quantum mechanics.[1]
More generally, we will encounter nonlinear operators.The simplest example is the different energy functional in various function spaces.Nonlinear operators play an important role in differential geometry and differential equation theory. For example, the minimal surface is the minimum point of the energy functional.[1]
Features and contents
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The characteristic of functional analysis is that it not only generalizes the basic concepts and methods of classical analysis, but also geometricalizes them.For example, different types of functions can be regarded as“function space ”Finally, the general concept of "abstract space" is obtained.It includes not only the geometric objects discussed previously, but also different function spaces.
Functional analysis is a powerful tool for studying modern physics.N-dimensional space can be used to describefreedomThe movement of the mechanical system ofMathematical toolsTo describe a mechanical system with infinite degrees of freedom.For example, the vibration of a beam is an example of an infinite number of degrees of freedom mechanical system.Generally speaking, fromparticleThe transition from mechanics to continuum mechanics requires a transition from a finite degree of freedom system to an infinite degree of freedom system.In modern physicsQuantum fieldTheory belongs to infinite degree of freedom system.
As the study of finite degree of freedom systems requires the geometry andCalculusAs a tool, the study of systems with infinite degrees of freedom requires the geometry and analysis of infinite dimensional space, which is the basic content of functional analysis.Therefore, functional analysis can also be popularly called infiniteDimensional spaceGeometry and calculus.The basic method in classical analysis is to uselinearIt can be applied to functional analysis to approximate nonlinear objects.
Functional analysis is the most "young" branch of analytical mathematics. It is the extension of classical analysis. It combines the views of function theory, geometry and algebra to study functions, operators, andLimit theory。In the 1940s and 1950s, he had become a mathematical subject with complete theory and rich contents.
For more than half a century, functional analysis, on the one hand, has used materials provided by many other disciplines to extract its own research objects and some research methods, and has formed many important branches of its own, such asOperator spectrum theory、Banach algebra, topological linear space theoryGeneralized function theorywait;On the other hand, it also strongly promotes the development of many other analytical disciplines.It is used in differential equations, probability theory, function theoryContinuum mechanics、Quantum physics、Computational mathematics、cybernetics、optimization theory There are important applications in such disciplines asCluster Upregulation AnalysisThe basic tool of theory is also to study infinite degrees of freedomphysical system One of the important and natural tools of.Today, its views and methods have infiltrated into many engineering disciplines and become one of the foundations of modern analysis.
Functional analysis in mathematical and physical equationsprobability theory、Computational mathematics、Continuum mechanics、Quantum physicsAnd other disciplines have a wide range of applications.In the past decade, functional analysis has gained more effective application in engineering technology.It also penetrates into various branches of mathematics and plays an important role.[3]
