Functional analysis (mathematics). Functional analysis

Function is one of the most important concepts of modern TRIZ. The function model is a triad: the subject (carrier) of the function, the action, the object of the function. An action can be expressed as an action verb or a parameter and the direction of its change. For example, the flame increases the temperature of the stove, the flame reduces the weight hot air balloon, the liquid near the phase transition stabilizes the temperature of the object, etc. An example of the formulation of the objective function: the pencil changes the color (colors) of the paper; the decoder restores the original message; The debug interface writes the necessary information to a log file.

To conduct a functional analysis, you need to know and understand several terms, which we list below.

The bearer of a function is the subject who implements the function in question.

Function object – the object to which the action of the function in question is directed.

A useful function is a function that determines the consumer properties of an object.

A harmful function is a function that negatively affects the consumer properties of an object.

Neutral function is a function that does not affect changes in the consumer properties of the object.

The main function is a useful function that reflects the purpose of the object (the purpose of its creation).

An additional function is a useful function that, together with the main function, ensures the manifestation of the consumer properties of an object.

The main function is a function that ensures the execution of the main one.

An auxiliary function of the first rank is a function that ensures the execution of the main one.

An auxiliary function of the second rank is a function that ensures the execution of an auxiliary function of the first rank. Auxiliary functions of the third and other lower ranks are functions that are subordinate to the functions of the previous rank.

Function rank is the significance of a function, determining its place in the hierarchy of functions that ensure the execution of the main function.

The level of performance of a function is the quality of its implementation, characterized by the value of the parameters of the function carrier.

Required parameters – parameters corresponding to the real operating conditions of the object.

Actual parameters are parameters inherent in the analyzed object (existing or designed).

An adequate level of function performance is the correspondence of actual parameters to the required ones.

Excessive level of function performance - an excess of actual parameters over the required ones.

Insufficient level of function performance – excess of required parameters over actual ones.

A component model is a model that reflects the composition of an object and the hierarchy (subordination) of its elements.

A structural model is a model that reflects the relationships between the elements of an object. Creating component and structural models is called component-structural analysis.

A functional model is a model that reflects the complex of functions of the object of analysis and its elements.

Functionally ideal model – functional model after applying convolution, reflecting the complex of object functions implemented by a minimum number of elements.

An undesirable effect is a defect in an object identified during the analysis process.

In some cases, when there is no directed action, but only the interaction of objects, the subject of the function cannot be distinguished from the object of the function. For example, when two radioactive substances interact, a nuclear reaction can be activated and an explosion occurs. In this case, both substances act on each other. In optics, when creating mirrors on a grinding machine, at some stage it is possible to produce two mirrors at once: convex and concave. Both glasses process each other.

The concept of function in TRIZ is most closely related to the concept “ parameter" The parameter has several important characteristics:

– the parameter does not exist on its own, it is always tied to one or another object, characterizes the state of this object;

– you can change the value of a parameter only by influencing the object;

– time is a parameter for processes or operations;

– the parameter can be measured in one way or another, including expert assessments;

– for the same parameter there are at least two objects characterized by this parameter; the parameter cannot be unique only for one system;

– the parameter can be increased, decreased, stabilized, controlled, compared;

– object parameters can be interrelated;

– the mutual relationship (dependence) between the parameters of an object is determined by the properties of this object;

– an object can be characterized different parameters depending on the aspect of its consideration.

Object parameters can be connected by cause-and-effect chains and create hierarchical parametric structures.

We can distinguish material and intangible aspects of considering the system.

Material aspects:

– physical (micro and macro)

– chemical

– biological

– technical

– art (material component).

Intangible aspects:

– psychological

– aesthetic and artistic

– social (individual, group, public, behavioral)

– organizational and structural

– business (business model, methods and technologies of doing business)

– personal-psychological

– linguistic

– financial and economic

– legal

– political

– research

– abstract mathematical (sets, programs, formal logic, etc.)

Depending on the aspect of consideration of the system, the parameters can be:

– informational (data transfer speed, reliability, security, etc.),

– technical (performance, reliability, measurement accuracy, etc.),

– economic (profit, liquidity, profitability, etc.),

– physical (temperature, mass, pressure, illumination, etc.),

– biochemical (glucose level, cholesterol level, antibody titer, etc.), etc.

Highly specialized parameters can also be used. For example, for hard magnetic disks (hard drives) special parameters are used: disk diameter, number of sectors per track, data transfer speed, transition time from one track to another, etc.

Some parameters, for example, informational ones, can be formed as a result of the state of other parameters, for example, technical, physical, chemical, biological.

The effectiveness of all functional analysis depends on the quality of the formulation of function models. There is a danger of making two fundamental mistakes. The first is to formulate actions in the form of a verb that does not actually describe the action. For example, love, work, toil, correct - such verbs will not help describe the action. We need a specific parameter that changes as a result of this action. The second one is pretty typical mistake– incorrect or inaccurate formulation of the subject or object of the function. For example, it is often forgotten that the main function object is outside the system in question. For example, text editors are aimed at interacting with a user who is not itself part of the editor. When formulating functions for immaterial systems, these problems of function formulation only become more acute. For example, in information technology The object of a function and the subject of a function very often change places in time. Thus, when working with a database, the user is either a provider of information or a consumer of information.

An example of a functional model of a software product was given in section 2.3.2. To build a functional model, you must first build a component model (what the system consists of). This is also useful from the point of view of finding resources to solve the problem. Then a structural model is built - which elements are connected to each other in the system and which are not. After this, for each component (element of the system), a function or several functions are formulated and a functional model of the system is constructed, on the basis of which a functional analysis is carried out.

The constructed functional model of the system allows, in particular, to carry out cause-and-effect analysis, highlighting the main shortcomings existing in the system and building cause-and-effect chains to determine the causes of the main shortcomings. This allows us to formulate the key shortcomings of the system, the solution of which, like the principle of falling dominoes, leads to the elimination of a whole group of shortcomings.

One of the options for functional analysis is functional cost analysis(FSA). It can be described in a simplified way as follows. Each element is assigned a specific function or set of functions, and their significance for the system as a whole is determined. After this, total costs are determined for the same components (elements). The distribution of the functional importance of elements is compared with the distribution of costs for that element. Those elements that have high costs, are associated with a large number of undesirable elements and at the same time have a low functional rank - these are the first candidates for folding in this system.

As an example, a simplified diagram comparing functional significance and cost level for task 7 is shown. From the diagram it can be seen that for the recognition block and the verification block the relationship between functional significance and costs is the worst. These blocks need to be collapsed (removed) first (Fig. 2.17).

Carrying out a deep functional analysis with setting tasks for collapsing is an independent section of TRIZ that requires a more in-depth study and additional tools analyzing the situation and setting tasks.

Another analytical tool is stream analysis tool(analysis of the flows of energy, matter and information available in the system). Using this analytical tool, deficiencies can be identified, tasks can be formulated, or the reasons for their occurrence can be identified.

Cause-and-effect analysis (CAA) is based on building cause-and-effect chains of existing shortcomings in the system. This chain can be constructed in the form of a graphical or other model reflecting the interdependence of the system's shortcomings.

Method “Accept the unacceptable”" - another method of analysis problematic situation and finding its solution. Its essence is to assume such a change in the system that under no circumstances is allowed under the conditions of the problem. Having allowed such an “unacceptable” change, a cause-and-effect chain is then built: what changes occur in the system, can they remove the prohibitions that prevented us from making this change?

The simplest examples of using the “Accept the Unacceptable” method can be taken from the experience of creating presentations. An obvious limitation: the width of a text block on a slide cannot be increased so that this block “climbs” onto the image surrounding it. Let's do the best we can and still increase the width of this text block. Quite often, the height of the text block is automatically reduced, and increasing its width no longer leads to “climbing” into the surrounding image.

To analyze situations and set tasks in TRIZ they often use sabotage analysis. Main idea sabotage analysis is that instead of solving a problem, the question is raised about how a problem can be created. There are two areas of application of diversion analysis in TRIZ. The first is how to explain the occurrence of this or that phenomenon. To do this, the task is set: how to create this phenomenon using only the available resources of the system. Secondly, the problem is set about how the system could be corrupted. This can be done sequentially by reversing all useful functions of the system. For example, in a sorting program you need to make sure that the elements of an array are moved to the wrong place. Knowing this, you can avoid mistakes when creating a program.

Page 1

Functional analysis is close to causal analysis, which is associated with great difficulties. To paraphrase Bacon, there are cases in which A precedes B and all modifications of A are accompanied by modifications of B, and the other variables are constant.  

Twenty years have passed since functional analysis in normed spaces.  

Functional analysis is a relatively recently emerged scientific discipline. As an independent branch of mathematical analysis, it took shape only in the last twenty to thirty years, which did not prevent it, however, from occupying one of the central places in modern mathematics.  

Functional analysis considers suitably chosen classes of functions as sets of points in topological spaces (chapter. Elegant and rich in geometric analogies, the derivations of the theory of linear transformations introduced in chapter. Solutions of linear differential equations, ordinary and partial derivatives, and linear integral equations are found by more or less simple generalization of solutions to systems linear equations, in particular, eigenvalue problems can be included here (Sec.  

Functional analysis consists of analyzing the product for each output function. possible reasons its violations, gradually reaching a given level of disaggregation. In this case, it is possible to identify failures that have the same external manifestations.  

Functional analysis is a set of physical and chemical methods analysis, using which can be qualitatively and quantitatively determined in organic compounds reactive groups of atoms (or individual atoms), so-called functional groups.  

Functional analysis is subordinated to the main task - the preliminary determination of parameters for given quality indicators based on consideration of the physical principle of the product and a rational technical solution. In the construction of mathematical models of functioning, the main attention is paid to the methodology for applying the methods of functional analysis. They try to apply the methods of functional analysis in their purest, simplest and most fundamental form.  

Functional analysis has great value for identification, since it allows one to establish the type of an unknown compound, its molecular weight or some part of it, as well as the ratio of functional groups.  

Functional analysis is not always complete strict decision, since the main purpose may be to develop a basic mathematical model of operation. The development of a basic model allows us to delve deeper into the problem and more fully understand the physical laws and assumptions made. It is especially preferable when solving new problems, and in many cases they are satisfied with an approximate estimate of the values ​​of quantities essential to the problem, and do not look for ways to determine them accurately. Sometimes finding such paths is very difficult or even impossible. Comparison of approximate values ​​of various parameters in the basic model often creates the basis for constructing a correct picture of the development of the process, for highlighting the main thing in it and discarding minor details. Most real problems of functional analysis when constructing a basic mathematical model of functioning are best solved using a generalized approach, and especially when a formal approach is completely unacceptable. In the generalized approach, due to the presence of several functional properties, the method of similarity theory and the method of dimensions are used.  

Functional analysis involves determining the type of functional group (e.g., aldehyde, carbonate, or hydroxyl) present in the sample being tested, without specifying which specific compound contains the functional group. Sometimes this information is not sufficient to accurately identify a compound if, for example, it can exist in the form of several isomers. Thus, the [P MH3Sb] complex, as has already been shown (Chapter IV), can be presented in the form of a trans or r isomer. Accurately identifying the isomer that is present in a system is a very difficult task, requiring the use of more specialized chemical and physical methods. Problems of this kind are very often encountered in the analysis of complex and especially organic compounds.  

Functional analysis studies sets equipped with mutually consistent algebraic and topological structures and their mappings, as well as methods by which information about these structures is applied to specific problems.  

Functional analysis and computational mathematics - (atics.  

Functional analysis studies certain topological-algebraic structures, as well as methods by which information from these structures can be applied to analytical problems.  

Functional analysis plays important role in modern mathematical education of a research engineer who has to apply mathematical methods in a specific field of science. In the language of functional analysis, the main problems of applied and computational mathematics are clearly expressed.  

OK-1 mastery of a culture of thinking, the ability to generalize, analyze, perceive information, set a goal and choose ways to achieve it, the ability to construct oral and written speech in a logically correct, reasoned and clear manner

OK-2 readiness to cooperate with colleagues and work in a team; knowledge of the principles and methods of organizing and managing small teams; ability to find organizational and managerial solutions in non-standard situations and is ready to bear responsibility for them

PK-25 ability to justify the correctness of the chosen model by comparing the results of experimental data and the solutions obtained

PK-26 readiness to use mathematical methods of processing, analysis and synthesis of professional research results

1.3. Planned results of mastering the discipline

knowledge: on a functional approach to the study of problems of system analysis and management problems based on modern mathematical technologies, models and methods for representing mathematical models in function spaces with operators preserving these spaces, as well as applications of methods for solving complex scientific and engineering problems, methods for solving problems with proof of convergence of solutions;

skills: correctly and constructively select and apply methods for solving engineering and scientific problems using knowledge in the field of functional analysis, develop knowledge bases corresponding to methods and models of functional analysis, select and use application software packages.

skills: formalization of problems, selection and construction of the necessary types of algebraic structures and their use to solve intellectual problems.

2. The place of discipline in OOP

The discipline “Introduction to Functional Analysis”, according to the federal state educational standard, training direction 230400 “Information systems and technologies” (qualification: bachelor) is a discipline of the educational cycle ().

Studying the discipline “Introduction to Functional Analysis” is based on the results of mastering the following disciplines:

The results of studying the discipline “Introduction to Functional Analysis” are used in the study of the following disciplines:

3. Distribution of labor intensity of mastering a discipline by type academic work

3.1. Types of educational work

Types of educational work

Labor intensity

Page 2 of 7

01.04.2012 22:25

Laboratory exercises

Practical classes, seminars

including classroom lessons in an interactive form

Independent work

including creative problem-oriented independent

Exams (preparation during the session, passing)

The total complexity of mastering the discipline

in academic hours:

in credit units:

3.2. Forms of control

Forms of control

Quantity

Current control

Test work (CRab), pcs.

Colloquiums (Kk), pcs.

Calculated graphic works(RGR), pcs.

Abstracts (Ref), pcs.

Course projects (CP), pcs.

Coursework (CR), pcs.

Interim certification

Tests (Z), pcs.

Exams (E), pcs.

4.1. Sections of the discipline and types of academic work

Finite-dimensional Euclidean space

Infinite-dimensional Euclidean space

Metric spaces

Metric spaces

Continuous operators in metric spaces

Normed spaces

Normed spaces

Hilbert space

Hilbert space

L2 space

Page 3 out of 7

01.04.2012 22:25

Rich text editor, text_content, press ALT 0 for help.

http://license.stu.neva.ru/programs/print/printprograms_code.php

		L2 space
		Linear operators
		Linear operators
		Total by type of educational work, ac
		Total by type of educational work, z
		Total labor intensity of development, h/z
			Topics, sections				Development results
								disciplines
1. Finite-dimensional Euclidean space
1.1. Finite-dimensional Euclidean space							Knowledge and skills at the level
The concept of space in mathematics.							concepts, definitions,
vector		space. Vector norm.				Scalar	descriptions, wording,
product of vectors. Linear transformations.
Matrices. Norm of the linear transformation operator.
2. Infinite-dimensional Euclidean space
2.1. Infinite-dimensional Euclidean space							Knowledge and skills at the level
Vectors with an infinite set of coordinates.							concepts, definitions,
L2 space.			Convergence		sequences		descriptions, wording,
vectors.		Continuity				scalar
works. Linear				functionals. Linear
operators.
3. Metric spaces
3.1. Metric spaces							Knowledge and skills at the level
Concepts of set theory. Definition of metric							concepts, definitions,
space. Convergence in metric space.							descriptions, wording,
Closed and open sets. Full metric
space. Countable sets.

4. Continuous operators in metric spaces

4.1. Continuous operators in metricKnowledge and skills at the level

spaces				concepts, definitions,
Basic definitions. Continuous operators and descriptions, formulations,
functionals.	Fixed		Skills method.
successive approximations. Compression operators.
Integral equations. Peano's theorem.
5. Normed spaces
5.1. Normed spaces				Knowledge and skills at the level
Linear systems. Normalized spaces.				concepts, definitions,
Finite-dimensional spaces. Subspaces. Problem about				descriptions, wording,
best approximation. Spaces with
6. Hilbert space
6.1. Hilbert space				Knowledge and skills at the level
Dot product. Definition of Hilbertian				concepts, definitions,
Basic	properties	space		Scalar	concepts, definitions,
work.	Orthogonal				descriptions, wording,
successive approximations for the integral
Fredholm equations. Average value of the function.
8. Linear operators
8.1. Linear operators					Knowledge and skills at the level
Additive operators. Linear operators.					concepts, definitions,
Limitation	linear	operators.	Spreading		descriptions, wording,
linear operators. Space of linear operators.
Inverse operators. Matrix linear operators.

5. Educational technologies

The course is taught primarily using traditional educational technologies:

– lectures,

– practical exercises.

Classes in active and interactive forms

Not provided.

6. Laboratory workshop

"Not provided"

7. Practical exercises

The program provides the following practical classes:

1. Euclidean spaces.

2. Normalized spaces.

3. Hilbert spaces.

4. Linear operators.

5. Inverse operators.

8. Organization and educational and methodological support for students’ independent work

SRS is aimed at consolidating and deepening the mastery of educational material and developing practical skills. SRS includes the following types independent work of students:

Approximate distribution of students' independent work time.

Approximate type of independent work, labor intensity,

Current SRS

Approximate distribution of students' independent work time

Approximate type of independent work, labor intensity,

Current SRS

9. Educational and methodological support of the discipline

9.1. Course website address saiu.ftk.spbstu.ru

Basic literature

9.3. Technical means ensuring discipline

"not provided"

10. Logistics ensuring discipline

"not provided"

11. Assessment criteria and assessment tools

11.1. Evaluation criteria

1. To assess knowledge in the discipline in case of answers on tickets (two questions on the ticket):

1.1. If you answer one question correctly on the ticket, the score is “Satisfactory.”

1.2. In case of correct answer to two questions on the ticket plus one additional question on the topics of the summary, the grade is “Good”.

1.3. In case of correct answer to two questions on the ticket and answer to two or more, at the teacher’s discretion, additional questions on the topics of the summary - the grade is “Excellent”.

1.4. In other cases - the rating is "Unsatisfactory".

11.2. Evaluation tools

List of exam questions for the discipline "Introduction to functional

Page 6 out of 7

01.04.2012 22:25

Rich text editor, text_content, press ALT 0 for help.

http://license.stu.neva.ru/programs/print/printprograms_code.php

1. The concept of space in mathematics. n-dimensional vector space.

2. Vector norm. Dot product of vectors.

3. Linear transformations. Matrices.

4. Norm of the linear transformation operator.

5. Vectors with an infinite set of coordinates.

6. L2 space. Convergence of a sequence of vectors.

7. Continuity of norm and scalar product.

8. Linear functionals. Linear operators.

9. Concepts of set theory.

10. Definition of metric space.

11. Convergence in metric space.

12. Closed and open sets.

13. Complete metric spaces. Countable sets.

14. Continuous operators and functionals.

15. Fixed points. Method of successive approximations.

16. Compression operators. Integral equations.

17. Peano's theorem.

18. Linear systems. Normalized spaces.

19. Finite-dimensional spaces. Subspaces.

20. Best approximation problem.

21. Spaces with a countable basis.

22. Dot product. Definition of Hilbert space.

23. The concept of orthogonality. Projection of an element onto a subspace.

24. Orthogonal decompositions of Hilbert space. Orthogonal systems of elements.

25. Basic properties of the L2 space.

26. Orthogonal series. Method of successive approximations for the Fredholm integral equation.

27. Average value of the function.

28. Additive operators.

29 Limitation of linear operators. Propagation of linear operators.

30. Space of linear operators.

31. Inverse operators. Matrix linear operators.

Feature educational process in the discipline "Introduction to Functional Analysis" is the need for students to receive a significant part necessary information when using educational and methodological and reference books in the process of independent work on practical assignments.

and linear mappings. It is characterized by a combination of methods of classical analysis, topology and algebra. Abstracting from specific situations, it is possible to identify axioms and, on their basis, construct theories that include classical problems such as special case and providing the opportunity to solve new problems. The process of abstraction itself has an independent meaning, clarifying the situation, discarding unnecessary things and revealing unexpected connections. As a result, it is possible to penetrate deeper into the essence of mathematical concepts and pave new paths of research.

Development Functional analysis (mathematical) occurred in parallel with the development of modern theoretical physics, and it turned out that the language Functional analysis (mathematical) most adequately reflects the laws of quantum mechanics, quantum field theory, etc. In turn, these physical theories had a significant impact on the problems and methods Functional analysis (mathematical)

1. The emergence of functional analysis. Functional analysis (mathematical) as an independent branch of mathematics developed at the turn of the 19th and 20th centuries. Large role in the formation of general concepts Functional analysis (mathematical) played by created G. Cantor set theory. The development of this theory, as well as axiomatic geometry, led to the emergence in the works of M. Frechet and F. Hausdorff metric and more general so-called. set-theoretic topology, which studies abstract spaces, i.e., sets of arbitrary elements for which the concept of proximity is established in one way or another.

Among abstract spaces for mathematical analysis and Functional analysis (mathematical) Functional spaces (that is, spaces whose elements are functions) turned out to be important - hence the name “ Functional analysis (mathematical)"). In the works of D. Gilbert By deepening the theory of integral equations, spaces arose l 2 And L 2(a, b) (see below). Generalizing these spaces, F. Rice studied spaces l p And Lp(a, b), and S. Banach in 1922 he identified complete linear normed spaces (Banach spaces). In the 1930-40s. in the works of T. Carleman , F. Rees, American mathematicians M. Stone and J. Neumann an abstract theory of self-adjoint operators in Hilbert space was constructed.

In the USSR, the first studies on Functional analysis (mathematical) appeared in the 30s: works

A.N. Kolmogorov (1934) on the theory of linear topological spaces;

N.N. Bogolyubova (1936) on invariant measures in dynamic systems;

L.V. Kantorovich (1937) and his students on the theory of semi-ordered spaces, applications Functional analysis (mathematical) to computational mathematics, etc.; M. G. Kerin and his students (1938) on an in-depth study of the geometry of Banach spaces, convex sets and cones in them, the theory of operators and connections with various problems of classical mathematical analysis, etc.; I.M. Gelfand and his students (1940) on the theory of normed rings (Banach algebras), etc.

For modern stage development Functional analysis (mathematical) characterized by strengthening connections with theoretical physics, as well as with various sections of classical analysis and algebra, for example, the theory of functions of many complex variables, the theory of partial differential equations, etc.

2. The concept of space. The most common spaces appearing in Functional analysis (mathematical), are linear (vector) topological spaces, i.e. linear spaces X over the field of complex numbers (or real numbers), which are also topological, and linear operations are continuous in the topology under consideration. A more particular, but very important situation arises when in linear space X one can introduce a norm (length) of vectors, the properties of which are a generalization of the properties of the length of vectors in ordinary Euclidean space. Namely, the norm of the element x Î X called real number ||x|| such that always || x|| ³ 0 and || x|| = 0 if and only if x = 0;

||l x|| = |l| || x||, l О x, if || x n - x|| 0.

In a large number of problems, an even more particular situation arises when in a linear space X we can introduce a scalar product - a generalization of the usual scalar product in Euclidean space. Namely, the scalar product of the elements x, at Î X is called a complex number ( x, at) such that always ( x, x) ³ 0 and ( x, x) = 0 if and only if x = 0;

, l, m О is the norm of the element x. Such a space is called pre-Hilbert. For designs Functional analysis (mathematical) it is important that the spaces under consideration are complete (i.e., from the fact that For x m, x n Î X, follows the existence of a limit, which is also an element X). A complete linear normed space and a complete pre-Hilbert space are called Banach and Hilbert spaces, respectively. In this case, the well-known procedure for completing the metric space (similar to the transition from rational numbers to real ones) in the case of a linear normed (pre-Hilbert) space leads to a Banach (Hilbert) space.

Normal Euclidean space is one of the simplest examples (of a real one) Hilbert space . However, in Functional analysis (mathematical) the main role is played by infinite-dimensional spaces, i.e. those in which there is an infinite number of linearly independent vectors. Here are examples of such spaces, the elements of which are complex-valued classes (that is, with values ​​in , norm || x|| = ; Banach space Lp(T) all summed with r th ( p³ 1) degree of functions on T, norm ; Banach space l p of all sequences such that , here (to the set of integers), norm || x|| =(å| x j|p) 1/ p ; in case p= 2 spaces l 2 And L 2 (T) are Hilbertian, and, for example, in L 2(T) dot product ; linear topological space D(), consisting of infinitely differentiable functions on , each of which is finite [i.e. i.e. equal to zero outside a certain interval ( A, b)]; at the same time x n x, If x n(t) are uniformly finite [i.e. e. ( A, b) does not depend on n] and converge uniformly with all their derivatives to the corresponding derivatives x(t).

All these spaces are infinite-dimensional, this is most easily seen for l 2: vectors e j= (0,..., 0, 1, 0,...) are linearly independent.

From a geometric point of view, the simplest are Hilbert spaces N, whose properties most closely resemble the properties of finite-dimensional Euclidean spaces. In particular, two vectors x, at Î N are called orthogonal ( x ^ y), If ( x, at) = 0. For any x Î N there is its projection onto an arbitrary subspace - linear closed subset N, i.e. such a vector x, What x-x^f for anyone f Î . Thanks to this fact large number geometric structures that take place in Euclidean space are transferred to N, where they often acquire an analytical character. For example, the usual orthogonalization procedure leads to the existence in N orthonormal basis - sequence of vectors e j, jО , from N such that || e j|| = 1, e j ^ e k at j ¹ k, and for anyone x Î the “coordinatewise” expansion is valid

x = å x j e j (1)

Where x j = (x, e j), ||x|| = å| x j| 2 (for simplicity N is assumed to be separable, i.e., it contains a countable everywhere dense set). If as N take L 2 (0, 2p) and put , j=...,-1, 0, 1..., then (1) will give the expansion of the function x(t) Î L 2 (0, 2p) into a Fourier series convergent in the mean square. In addition, relation (1) shows that the correspondence between N And l 2 " (xj), jО Hilbert spaces Hj- education-like construction N one-dimensional subspaces described by formula (1); factorization and completion: on the original linear space X a quasi-scalar product is specified [i.e. i.e. equality is possible ( x, x) = 0 for x¹ 0], often of a very exotic nature, and N is built by the replenishment procedure X relative to (.,.) after preliminary identification with 0 vectors x, for which ( x, x) = 0; tensor product - its formation is similar to the transition from functions of one variable f(x 1) to functions of many variables f(x 1,..., x q); the projective limit of Banach spaces is here (roughly speaking) if for each a; inductive limit of Banach spaces X 1 Ì X 2 M..., here if everything x j, starting from some j 0, lie in one X j0, and in it . The last two procedures are usually used to construct linear topological spaces. These are, for example, nuclear spaces - the projective limit of Hilbert spaces N a having the property that for every a there is a b such that h b М N a, and this is the so-called Hilbert–Schmidt embedding.

An important section of F, a., has been developed, in which spaces with a conical structure are studied "x 0" (semi-ordered). An example of such a space is real WITH(T), it is considered x 0 if x(t³)0 for all t Î T.

3. Operators (general concepts). Functionalities. Let X, - linear spaces; display A: X ® is called linear if for x, at Î X, l, m О ,

Where x 1,..., x n And ( Ax) 1 ,..., (Ax) n- vector coordinates x And Ax respectively. When passing to infinite-dimensional linear topological spaces, the situation becomes significantly more complicated. Here, first of all, it is necessary to distinguish between continuous and discontinuous linear operators (for finite-dimensional spaces they are always continuous). Thus, acting from space L 2 (A, b) into it the operator

(2)

(Where (t, s) - limited function - core A) - continuous, while defined on a subspace 1 (a, b) Ì L 2(a, b) differentiation operator

(3)

is discontinuous (in general, characteristic feature discontinuous operators is that they are not defined over the entire space).

Continuous operator A: X ® , Where X, - Banach spaces, is characterized by the fact that

,

therefore it is also called limited. The set of all restricted operators ( X, ) with respect to ordinary algebraic operations forms a Banach space with norm || A||. Properties if for each x Î X], relative to which the ball, i.e., the set of points x Î X such that || x|| £ r, will already be compact (this effect will never happen in an infinite-dimensional space with respect to the topology generated by the norm). This allows us to study in more detail a number of geometric issues for sets of X", for example, to establish the structure of an arbitrary compact convex set as a closed shell of its extreme points (Krein-Milman theorem).

An important task Functional analysis (mathematical) is to find a general form of functionals for specific spaces. In a number of cases (besides the Hilbert space) this can be done, for example ( l p)¢, p> 1, consists of functions of the form å x j e j, Where , . However, for most Banach (and especially linear topological) spaces, functionals will be elements of a new nature that cannot be constructed simply by means of classical analysis. So, for example, for fixed t 0 And m in space D() functionality defined . In case m= 0 it can still be written in a “classical” way - using an integral, but with m³ 1 this is no longer possible. Elements from ( D())¢ are called generalized functions (distributions). Generalized functions as elements of the conjugate space can also be constructed when D() is replaced by another space Ф, consisting of both an infinitely and a finitely many times differentiable functions; in this case, an essential role is played by triples of spaces Ф" É NÉ F, where N is the original Hilbert space, and Ф is a linear topological (in particular, Hilbert with other scalar product) space, for example

F = l 2(T).

Differential operator D, appearing in (3), will be continuous if understood as acting in L 2[a, b] from space 1 [a, b], supplied with the norm However, for many problems, and primarily for spectral theory, such differential operators must be interpreted as acting in the same space. These and other related problems led to the construction general theory unbounded, in particular unbounded self-adjoint, and Hermitian operators.

4. Special classes of operators. Spectral theory. Many problems lead to the need to study the solvability of an equation of the form Cx = y, Where WITH- some operator, at Î - given, and x Î X- the desired vectors. For example, if X = = L 2 (A, b), WITH = E - A, Where A is the operator from (2), and E is the identity operator, then we obtain the Fredholm integral equation of the 2nd kind; If WITH is a differential operator, then a differential equation is obtained, etc. However, here one cannot count on a fairly complete analogy with linear algebra without limiting the class of operators under consideration. One of the most important classes of operators that are closest to the finite-dimensional case are compact (completely continuous) operators, characterized by the fact that they take each bounded set from X to many of , whose closure is compact [such as, for example, the operator A from (2)]. For compact operators, a theory of solvability of the equation is constructed x - Ax = at, quite analogous to the finite-dimensional case (and containing, in particular, the theory of the mentioned integral equations) (F. Ries).

In a variety of tasks mathematical physics the so-called task for eigenvalues : for some operator A: X ® X it is required to find out the possibility of finding a solution j ¹ 0 ( eigenvector ) equations A j = lj for some l О l j x j e j, (4)

where l j, - eigenvalue corresponding e j. For finite-dimensional X the question of such a representation is completely clarified, and in the case of multiple eigenvalues ​​to obtain a basis in X you need, generally speaking, to add to your own so-called. associated vectors. The set SpA of eigenvalues ​​in this case is called a spectrum A.

The first transfer of this picture to the infinite-dimensional case was given for integral operators of the type A from (2) with a symmetric kernel [i.e. e. K(t, s) = K(s, t) and indeed] (D. Gilbert). Then a similar theory was developed for general compact self-adjoint operators in Hilbert space. However, when moving to the simplest non-compact operators, difficulties arose related to. by the very definition of spectrum. Thus, the limited operator in L 2[a, b]

(Tx)(t) = tx(t) (5)

has no eigenvalues. Therefore, the definition of spectrum has been revised, generalized and now looks like this.

Let X- Banach space, AО is a polynomial, then f(A) = (the degree of an operator is understood as its sequential application). However, if f(z) is an analytical function, then it is straightforward to understand f(A) is no longer always possible; in this case f(A) is determined by the following formula if f(z) is analytic in a neighborhood of SpA, and Г is a contour enclosing SpA and lying in the domain of analyticity f(z):

. (6)

In this case, algebraic operations on functions transform into similar operations on operators [i.e. e. display f(z) ® f(A) - homomorphism]. These constructions do not make it possible to clarify, for example, issues of completeness of eigenvectors and associated vectors for common operators, however, for self-adjoint operators, which are of primary interest, for example, for quantum mechanics, such a theory is fully developed.

Let N- Hilbert space. Limited operator A: N ® N is called self-adjoint if ( Ax, at) = (x, Ay) (in case of unlimited A definition is more difficult). If N n-dimensional, then it contains an orthonormal basis of eigenvectors of the self-adjoint operator A; in other words, the expansions take place:

, , (7)

Where (l j) - projection operator (projector) onto a subspace spanned by all eigenvectors of the operator A, corresponding to the same eigenvalue l j.

It turns out that these formulas can be generalized to an arbitrary self-adjoint operator from N, only the projectors themselves (l j) may not exist, since there may be no eigenvectors [such, for example, the operator T in (5)]. In formulas (7), the sums are now replaced by Stieltjes integrals over a non-decreasing operator-valued function E(l) [which in the finite-dimensional case is equal to ], is called the expansion of unity, or the spectral (projector) measure, the growth points of which coincide with the spectrum Sp A. If we use generalized functions, then formulas like (7) are preserved. Namely, if there is a triple Φ" É N É F, where Ф, for example, is nuclear, and A transforms Ф into Ф¢ and continuously, then relations (7) hold, only the sums go into integrals over some scalar measure, and E(l) now “projects” Φ into Φ¢, giving vectors from Φ¢ that will be eigenvectors in the generalized sense for A with eigenvalue l. Similar results are valid for the so-called. normal operators (i.e., commuting with their conjugates). For example, they are true for unitary operators - such limited operators that display everything N for everything N and preserve the scalar product. For them the spectrum Sp located on a circle | z| = 1, along which integration is carried out in analogues of formulas (6). See also Spectral analysis linear operators.

5. Nonlinear functional analysis. Simultaneously with the development and deepening of the concept of space, there was a development and generalization of the concept of function. Ultimately, it turned out to be necessary to consider mappings (not necessarily linear) of one space into another (often into the original one). One of the central problems of nonlinear Functional analysis (mathematical) is the study of such mappings. As in the linear case, the mapping of space into ) is called a functional. For nonlinear mappings (in particular, nonlinear functionals), the differential, directional derivative, etc. can be determined in various ways. similar to the corresponding concepts of classical analysis. Selection from quadratic display, etc. terms leads to a formula similar to Taylor's formula.

An important task of nonlinear Functional analysis (mathematical) is the task of finding fixed points of the mapping (point x called stationary for display , If Fx = x). Many problems on the solvability of operator equations, as well as the problem of finding eigenvalues ​​and eigenvectors of nonlinear operators, come down to finding fixed points. When solving equations with nonlinear operators containing a parameter, an important problem arises for the nonlinear Functional analysis (mathematical) phenomenon - so-called branching points (decisions).

When studying fixed points and branch points, topological methods are used: generalizations to infinite-dimensional spaces of Brouwer's theorem on the existence of fixed points of mappings of finite-dimensional spaces, degrees of mappings, etc. Topological methods Functional analysis (mathematical) developed by the Polish mathematician J. Schauder, the French mathematician J. Leray, the Soviet mathematicians M. A. Krasnoselsky, L. A. Lyusternik and others.

6. Banach algebras. Representation theory. In the early stages of development Functional analysis (mathematical) problems were studied for the formulation and solution of which only linear operations on elements of space were necessary. The only exceptions are, perhaps, the theory of operator (factor) rings (J. Neumann, 1929) and the theory of absolutely convergent Fourier series (N. Wiener , 1936). At the end of the 30s. In the works of the Japanese mathematician M. Nagumo, Soviet mathematicians I. M. Gelfand, G. E. Shilov, M. A. Naimark and others, the theory of the so-called. normed rings (the modern name is Banach algebras), in which, in addition to linear space operations, the multiplication operation is axiomatized (and || xy|| £ || x|| ||y||). Typical representatives Banach algebras are rings of bounded operators acting in a Banach space X(multiplication in it - sequential application of operators - is necessary taking into account the order), various kinds functional spaces, for example c is the Haar measure on a group of characters, and

,

Generalized Fourier transform of functions f(g) And k(g), which continues to isomorphism L 2(G, dg) V L 2(, dc). For non-commutative groups the situation becomes much more complicated. If G is compact, then the representation of the group of shift operators (or, in short, the group of shifts) can be well described; in this case L 2(G, dg) decomposes into a direct sum of finite-dimensional translation-invariant subspaces. If G is non-compact, then we also obtain the decomposition L 2(G, dg) into simpler invariant parts, but not into a direct sum, but into a direct integral.

If G= , then the theory of unitary representations can be reduced to the theory of self-adjoint operators. Namely, the one-parameter group of unitary operators T l , l О in Hilbert space N admits representation T l = exp i l A, Where A- self-adjoint operator (Stone's theorem); operator A is called the infinitesimal operator (generator) of the group ( T" l). This result finds important applications in the study of phase space transformations of classical mechanics. This connection, as well as applications in statistical physics, underlies the extensive branch Functional analysis (mathematical) - ergodic theory . The connection between one-parameter transformation groups and their generators allows for significant generalizations: operators T l do not have to be unitary, can act in Banach and more general spaces, and even be defined only for l ³ 0 (the so-called theory of operator semigroups). This section Functional analysis (mathematical) has applications in the theory of partial differential equations and the theory of random (specifically Markov) processes.

Lit.: Lyusternik L.A., Sobolev V.I., Elements of functional analysis, 2nd ed., M., 1965; Kolmogorov A. N., Fomin S. V., Elements of the theory of functions and functional analysis 4th ed., M., 1976; Akhiezer N. I., Glazman I. M., Theory of linear operators in Hilbert space, 2nd ed., Moscow, 1966; Vulikh B.Z., Introduction to the theory of semi-ordered spaces, M., 1961; Banakh S.S., Course of functional analysis Kiev, 1948; Riess F., Szekefalvi-Nagy B., Lectures on functional analysis, trans. from French, M., 1954; Sobolev S.L., Some applications of functional analysis in mathematical physics, Leningrad, 1950; Kantorovich L.V., Akilov G.P., Functional analysis in normed spaces, M., 1959; Krasnoselsky M.A., Zabreiko P.P., Geometric methods of nonlinear analysis, M., 1975; Naimark M.A., Normalized rings, 2nd ed., M., 1968; Rudin U., Functional analysis, trans. from English, M., 1975; Yoshida K., Functional analysis, transl., from English, M., 1967; Dunford N., Schwartz J., Linear operators, trans. from English, parts 1-3, M., 1962-74; Hille E., Phillips R., Functional analysis and semigroups, trans. from English, 2nd ed., M., 1962; Edwards R. E., Functional analysis. Theory and applications translated from English, M., 1969.

Yu. M. Berezansky, B. M. Levitan.

Article about the word " Functional analysis (mathematical)" in the Great Soviet Encyclopedia was read 6486 times

I Functional analysis

part of modern mathematics, main task which is the study of infinite-dimensional spaces and their mappings. Linear spaces and linear mappings are the most studied. For F. a. Characterized by a combination of methods of classical analysis, topology and algebra. Abstracting from specific situations, it is possible to identify axioms and, on their basis, build theories that include classical problems as a special case and make it possible to solve new problems. The process of abstraction itself has an independent meaning, clarifying the situation, discarding unnecessary things and revealing unexpected connections. As a result, it is possible to penetrate deeper into the essence of mathematical concepts and pave new paths of research.

Development of F. a. occurred in parallel with the development of modern theoretical physics, and it turned out that the language of F. a. most adequately reflects the laws of quantum mechanics, quantum field theory, etc. In turn, these physical theories had a significant impact on the problems and methods of physical science.

1. The emergence of functional analysis. F. a. as an independent branch of mathematics developed at the turn of the 19th and 20th centuries. A major role in the formation of general concepts of F. a. played by the set theory created by G. Cantor. The development of this theory, as well as axiomatic geometry, led to the emergence in the works of M. Fréchet and F. Hausdorff of metric and more general so-called geometry. set-theoretic topology, which studies abstract spaces, i.e., sets of arbitrary elements for which the concept of proximity is established in one way or another.

Among abstract spaces for mathematical analysis and FA. Functional spaces turned out to be important (that is, spaces whose elements are functions - hence the name “F. a.”). In the works of D. Hilbert to deepen the theory of integral equations, spaces arose l 2 And L 2(a, b) (see below). Generalizing these spaces, F. Ries studied the spaces l p And Lp(a, b), and S. Banach in 1922 identified complete linear normed spaces (Banach spaces). In the 1930-40s. In the works of T. Carleman, F. Rees, and American mathematicians M. Stone and J. Neumann, an abstract theory of self-adjoint operators in Hilbert space was constructed.

In the USSR, the first studies on Ph. a. appeared in the 30s: works

A. N. Kolmogorov (1934) on the theory of linear topological spaces;

N. N. Bogolyubova (1936) on invariant measures in dynamical systems;

L. V. Kantorovich (1937) and his students on the theory of semi-ordered spaces, applications of f. a. to computational mathematics, etc.; M. G. Kerin and his students (1938) on an in-depth study of the geometry of Banach spaces, convex sets and cones in them, the theory of operators and connections with various problems of classical mathematical analysis, etc.; I. M. Gelfand and his students (1940) on the theory of normed rings (Banach algebras), etc.

For the current stage of development of F. a. characterized by strengthening connections with theoretical physics, as well as with various sections of classical analysis and algebra, for example, the theory of functions of many complex variables, the theory of partial differential equations, etc.

2. The concept of space. The most general spaces appearing in FA are linear (vector) topological spaces, that is, linear spaces (See Linear space) X over the field of complex numbers X, one can introduce a norm (length) of vectors, the properties of which are a generalization of the properties of the length of vectors in ordinary Euclidean space. Namely, the norm of the element x ∈ X is called a real number || x|| such that always || x|| ≥ 0 and || x|| = 0 if and only if x = 0;

||λ x || = |λ| || x||, λ ∈

||x + y|| ≤ ||x|| + ||y||.

Such a space is called linear normed; the topology in it is introduced using the dist metric ( x, at) = ||x - at|| (i.e. it is considered that the sequence x n x if || x n - x||

In a large number of problems, an even more particular situation arises when in a linear space X we can introduce a scalar product - a generalization of the usual scalar product in Euclidean space. Namely, the scalar product of the elements x, at ∈ X is called a complex number ( x, at) such that always ( x, x) ≥ 0 and ( x, x) = 0 if and only if x = 0;

At the same time x. Such a space is called pre-Hilbert. For F. a. designs. it is important that the spaces under consideration are complete (i.e., from the fact that xm, x n ∈ X, the existence of a limit X follows). A complete linear normed space and a complete pre-Hilbert space are called Banach and Hilbert spaces, respectively. Moreover, the well-known procedure for completing a metric space (similar to the transition from rational numbers to real numbers) in the case of a linear normed (pre-Hilbert) space leads to a Banach (Hilbert) space.

Ordinary Euclidean space is one of the simplest examples of a (real) Hilbert space (See Hilbert space). However, in F. a. the main role is played by infinite-dimensional spaces, i.e. those in which there is an infinite number of linearly independent vectors. Here are examples of such spaces, the elements of which are complex-valued classes (that is, with values ​​in x ( t), defined on some set T, with ordinary algebraic operations [i.e. e.(x + y)(t) = x(t) + y(t), (λ x)(t) = λ x(t)]

Banach space WITH(T) all continuous functions, T- compact subset n- dimensional space x|| = L p ( T) all summed with r th ( p≥ 1) degree of functions on T, norm l p of all sequences such that x|| =(∑x j | p) 1/p ; in case p= 2 spaces l 2 And L 2 (T) are Hilbertian, and, for example, in L 2(T) dot product D (|R), consisting of infinitely differentiable functions on |R, each of which is finite [i.e. i.e. equal to zero outside a certain interval ( A, b)]; at the same time x n x, if x n(t) are uniformly finite [i.e. e. ( A, b) does not depend on n] and converge uniformly with all their derivatives to the corresponding derivatives x(t).

All these spaces are infinite-dimensional, this is most easily seen for l 2: vectors e j= (0,..., 0, 1, 0,...) are linearly independent.

From a geometric point of view, the simplest are Hilbert spaces N, whose properties most closely resemble the properties of finite-dimensional Euclidean spaces. In particular, two vectors x, at ∈ N are called orthogonal ( x ⊥ y), If ( x, at) = 0. For any x ∈ N there is its projection onto an arbitrary subspace F- linear closed subset N, i.e. such a vector xF, What x-xF⊥f for anyone f ∈ F. Thanks to this fact, a large number of geometric structures that take place in Euclidean space are transferred to N, where they often acquire an analytical character. For example, the usual orthogonalization procedure leads to the existence in N orthonormal basis - sequence of vectors e j, j∈ H such that || e j|| = 1, e j ⊥ e k at j ≠ k, and for anyone x ∈ H the “coordinatewise” expansion is valid

x = ∑x j e j (1)

Where x j = (x, e j), ||x|| = ∑x j | 2 (for simplicity N is assumed to be separable, i.e., it contains a countable everywhere dense set). If as N take L 2 (0, 2π) and put j =...,-1, 0, 1..., then (1) gives the expansion of the function x(t) ∈ L 2 (0, 2π) into a Fourier series convergent in the mean square. In addition, relation (1) shows that the correspondence between N And l 2 ∋ (xj), j ∈

Such geometric questions become sharply more complicated when moving from Hilbert to Banach and, even more so, linear topological spaces due to the impossibility of orthogonal projection in them. For example, the “basis problem”. Vectors e, form a basis in l p in the sense of the validity of expansion (1). Most of the bases were built famous examples Banach spaces, however, the problem (S. Banach - J. Schauder) of the existence of a basis in every separable Banach space could not be solved for more than 50 years and only in 1972 was it solved negatively. In F. a. An important place is occupied by similar “geometric” topics devoted to elucidating the properties of various sets in Banach and other spaces, for example, convex, compact, etc. Here, simply formulated questions often have very non-trivial solutions. This topic is closely related to the study of isomorphism of spaces, with the discovery of universal (like l 2) representatives in a particular class of spaces, etc.

Large section of F. a. is devoted to a detailed study of specific spaces, because their properties usually determine the nature of the solution to the problem obtained by FA methods. Typical example- embedding theorems for the so-called. spaces of S. L. Sobolev and their generalizations: the simplest such space W l p(T), p ≥ 1, l= 0, 1, 2,..., is defined as the completion of the space of infinitely differentiable T functions x(t) relative to the norm ∑|| D α x|| V Lp(T), where the sum applies to all derivatives Dα to order ≤ l. These theorems clarify the question of the nature of the smoothness of space elements obtained by the completion procedure.

In connection with the requests of mathematical physics in Ph. A. A large number of specific spaces arose, built from previously known ones using certain structures. The most important of them:

orthogonal sum H j - formation-like construction N one-dimensional subspaces described by formula (1); factorization and completion: on the original linear space X a quasi-scalar product is specified [i.e. i.e. equality is possible ( x, x) = 0 for x≠ 0], often of a very exotic nature, and N is built by the replenishment procedure X relative to (.,.) after preliminary identification with 0 vectors x, for which ( x, x) = 0; tensor product f ( x 1) to functions of many variables f(x 1,..., x q); projective limit X 1 ⊂ X 2⊂..., here x j , starting from some j 0, lie in one X j0, and in it H α having the property that for every α there is a β such that h β ⊂ Nα, and this is the so-called Hilbert–Schmidt embedding.

An important section of F, a., has been developed, in which spaces with a conical structure are studied "x WITH ( T), it is considered x x( t≥)0 for all t ∈T.

3. Operators (general concepts). Functionalities. Let X, Y- linear spaces; display A: X → Y is called linear if for x, at ∈ X, λ, μ ∈

A(λ x + μ at) = λ Ax + μ Ay;

linear mappings are usually called linear operators. In the case of finite-dimensional X, Y the structure of a linear operator is simple: if we fix the bases in X And Y, That

Where x 1,..., x n And ( Ax) 1 ,..., (Ax) n- vector coordinates x And Ax respectively. When passing to infinite-dimensional linear topological spaces, the situation becomes significantly more complicated. Here, first of all, it is necessary to distinguish between continuous and discontinuous linear operators (for finite-dimensional spaces they are always continuous). Thus, acting from space L 2 (A, b) into it the operator

(Where K(t, s) - limited function - core A) - continuous, while defined on a subspace C 1(a, b) ⊂ L 2(a, b) differentiation operator

is discontinuous (in general, a characteristic feature of discontinuous operators is that they are not defined over the entire space).

Continuous operator A: X → Y, Where X, Y- Banach spaces, is characterized by the fact that

therefore it is also called limited. The set of all restricted operators X, Y) with respect to ordinary algebraic operations forms a Banach space with norm || A||. Properties X, Y) largely reflect the properties of the X And Y. This especially applies to the case when Y one-dimensionally, i.e. when linear continuous mappings are considered l: X→ X, X is a space and is denoted X". If X = N is Hilbertian, then the structure H" is simple: similar to the finite-dimensional case, each functional l(x) has the form ( x, a), Where a- depending on l vector from N(Ries' theorem). Correspondence H" → N establishes an isomorphism between H" And N, and we can assume that H" = N. In the case of a general Banach space X the situation is much more complicated: you can build X", X" = (X")" ,..., and these spaces may turn out to be different. In general, in the case of a Banach space, even the question of the existence of nontrivial (i.e., different from 0) functionals is not simple. If F- subspace X(not reducible to one point) and exists l ∈ F", then this functionality can be extended to anything X to functionality from X" without changing the norm (Hahn-Banach theorem). If l ∈ X, then the equation l(x) = c defines a hyperplane - a subspace shifted by a certain vector X, having one less than X, dimension, so that results like the theorem above have a simple geometric interpretation.

Space X" in a certain sense "better" X. So, for example, in it, along with the norm, you can introduce the so-called. weak topology [roughly speaking, x ∈ X], relative to which the ball, i.e., the set of points x ∈ X such that || x|| ≤ r, will already be compact (this effect will never happen in an infinite-dimensional space with respect to the topology generated by the norm). This allows us to study in more detail a number of geometric issues for sets of X", for example, to establish the structure of an arbitrary compact convex set as a closed shell of its extreme points (Krein-Milman theorem).

An important task of F. a. is to find a general form of functionals for specific spaces. In a number of cases (besides the Hilbert space) this can be done, for example ( l p)", p> 1, consists of functions of the form ∑ x j e j, where t 0 and m in space D(|R) the functional is defined m = 0, it can still be written in a “classical” way - using an integral, but with m≥ 1 this is no longer possible. Elements from ( D(|R))" are called generalized functions (See Generalized functions) (distributions). Generalized functions as elements of the dual space can also be constructed when D(|R) is replaced by another space Ф, consisting of both infinitely and finitely many times differentiable functions; in this case, an essential role is played by triples of spaces Ф" ⊃ N⊃ Ф, where N is the original Hilbert space, and Ф is a linear topological (in particular, Hilbert with other scalar product) space, for example

F = W l 2(T).

Differential operator D, appearing in (3), will be continuous if understood as acting in L 2[a, b] from space C 1[a, b], supplied with the norm

4. Special classes of operators. Spectral theory. Many problems lead to the need to study the solvability of an equation of the form Cx = y, Where WITH- some operator, at ∈ Y- given, and x ∈ X- the desired vectors. For example, if X = Y = L 2 (A, b), WITH = E - A, Where A is the operator from (2), and E is the identity operator, then we obtain the Fredholm integral equation of the 2nd kind; If WITH is a differential operator, then a differential equation is obtained, etc. However, here one cannot count on a fairly complete analogy with linear algebra without limiting the class of operators under consideration. One of the most important classes of operators that are closest to the finite-dimensional case are compact (completely continuous) operators, characterized by the fact that they take each bounded set from X to many of Y, whose closure is compact [such as, for example, the operator A from (2)]. For compact operators, a theory of solvability of the equation is constructed x - Ax = at, quite analogous to the finite-dimensional case (and containing, in particular, the theory of the mentioned integral equations) (F. Ries).

In various problems of mathematical physics, the so-called Eigenvalue problem: for some operator A: X → X it is required to find out the possibility of finding a solution φ ≠ 0 (eigenvector (See Eigenvectors)) of the equation Aφ = λφ for some λ ∈ A by an eigenvector is especially simple - it reduces to multiplication by a scalar. Therefore, if, for example, the eigenvectors of the operator A form a basis e j, j∈ X, i.e., there is a decomposition of type (1), then the action A becomes especially clear:

Ax= ∑ j x j e j , (4)

where λ j, - eigenvalue corresponding e j. For finite-dimensional X the question of such a representation is completely clarified, and in the case of multiple eigenvalues ​​to obtain a basis in X you need, generally speaking, to add to your own so-called. associated vectors. The set SpA of eigenvalues ​​in this case is called a spectrum A.

The first transfer of this picture to the infinite-dimensional case was given for integral operators of the type A from (2) with a symmetric kernel [i.e. e. K(t, s) = K(s, t) and indeed] (D. Gilbert). Then a similar theory was developed for general compact self-adjoint operators (See Self-adjoint operator) in Hilbert space. However, when moving to the simplest non-compact operators, difficulties arose related to. by the very definition of spectrum. Thus, the limited operator in L 2[a, b]

(Tx)(t) = tx(t) (5)

has no eigenvalues. Therefore, the definition of spectrum has been revised, generalized and now looks like this.

Let X- Banach space, A∈ X, X). Dot z∈ A if the inverse operator ( A - zE) –1 = Rz(i.e. the inverse mapping) exists and belongs to X, X). The complement to the set of regular points is called the spectrum Sp A operator A. As in the finite-dimensional case, Sp A is always non-empty and located in a circle || z|| ≤ ||A||. With the help of these concepts, the theory of Operators was built, i.e., it was found out how to give reasonable meaning to certain functions from operators. So, if f(z) = f ( A) = f ( z) is an analytical function, then it is straightforward to understand f(A) is no longer always possible; in this case f(A) is determined by the following formula if f(z) is analytic in a neighborhood of SpA, and Г is a contour enclosing SpA and lying in the domain of analyticity f(z):

In this case, algebraic operations on functions transform into similar operations on operators [i.e. e. display f(z) → f(A) - homomorphism]. These constructions do not make it possible to clarify, for example, questions of completeness of eigenvectors and associated vectors for general operators, however, for self-adjoint operators, which are of main interest, for example, for quantum mechanics, such a theory is fully developed.

Let N- Hilbert space. Limited operator A: N → N is called self-adjoint if ( Ax, at) = (x, Ay) (in case of unlimited A definition is more difficult). If N n-dimensional, then it contains an orthonormal basis of eigenvectors of the self-adjoint operator A; in other words, the expansions take place:

Where P(λ j) - projection operator (projector) onto a subspace spanned by all eigenvectors of the operator A, corresponding to the same eigenvalue λ j.

It turns out that these formulas can be generalized to an arbitrary self-adjoint operator from N, only the projectors themselves P(λ j) may not exist, since there may be no eigenvectors [such, for example, the operator T in (5)]. In formulas (7), the sums are now replaced by Stieltjes integrals over a non-decreasing operator-valued function E(λ) [which in the finite-dimensional case is equal to A. If we use generalized functions, then formulas like (7) are preserved. Namely, if there is a triple Ф" ⊃ N ⊃ F, where Ф, for example, is nuclear, and A takes Φ to Φ" and continuously, then relations (7) hold, only the sums go into integrals over some scalar measure, and E(λ) now “projects” Φ into Φ, giving vectors from Φ that will be eigenvectors in the generalized sense for A with eigenvalue λ. Similar results are valid for the so-called. normal operators (i.e., commuting with their conjugates). For example, they are true for unitary operators (See Unitary operator) U- such limited operators that display everything N for everything N and preserve the scalar product. For them the spectrum Sp U located on a circle | z| = 1, along which integration is carried out in analogues of formulas (6). See also Spectral analysis of linear operators.

5. Nonlinear functional analysis. Simultaneously with the development and deepening of the concept of space, there was a development and generalization of the concept of function. Ultimately, it turned out to be necessary to consider mappings (not necessarily linear) of one space into another (often into the original one). One of the central problems of nonlinear FA. is the study of such mappings. As in the linear case, mapping space into

An important task of nonlinear FA. is the task of finding fixed points of the mapping (point x called stationary for display F, If Fx = x). Many problems on the solvability of operator equations, as well as the problem of finding eigenvalues ​​and eigenvectors of nonlinear operators, come down to finding fixed points. When solving equations with nonlinear operators containing a parameter, something important for the nonlinear linear function arises. phenomenon - so-called branching points (decisions).

When studying fixed points and branch points, topological methods are used: generalizations to infinite-dimensional spaces of Brouwer's theorem on the existence of fixed points of mappings of finite-dimensional spaces, degrees of mappings, etc. Topological methods of F. a. developed by the Polish mathematician J. Schauder, the French mathematician J. Leray, the Soviet mathematicians M. A. Krasnoselsky, L. A. Lyusternik and others.

6. Banach algebras. Representation theory. In the early stages of development of F. a. problems were studied for the formulation and solution of which only linear operations on elements of space were necessary. The only exceptions are, perhaps, the theory of operator (factor) rings (J. Neumann, 1929) and the theory of absolutely convergent Fourier series (N. Wiener, 1936). At the end of the 30s. In the works of the Japanese mathematician M. Nagumo, Soviet mathematicians I. M. Gelfand, G. E. Shilov, M. A. Naimark and others, the theory of the so-called. normed rings (the modern name is Banach algebras), in which, in addition to linear space operations, the multiplication operation is axiomatized (and || xy|| ≤ ||x|| ||y||). Typical representatives of Banach algebras are rings of bounded operators acting in a Banach space X(multiplication in it - sequential application of operators - is necessary taking into account the order), various kinds of function spaces, for example C(T) with normal multiplication, L 1(|R) with convolution as a product, and their broad generalization is the class of so-called. group algebras (topological groups G), consisting of complex-valued functions or measures defined on G with convolution (in various, not necessarily equivalent, versions) Functional analysis of behavioral signs Many examples can be given when the function of a behavioral act is obvious and does not require special research. However, in many cases, painstaking research is required to find out, for example, the function of individual authors Krapivensky Solomon Eliazarovich

3. Social systems: functional analysis The analysis of social systems carried out above was primarily of a structural-component nature. For all its importance, it allows you to understand what the system consists of and - to a much lesser extent - what its purpose is.

9. Empirical sociology and structural-functional analysis

From the book Philosophy author Lavrinenko Vladimir Nikolaevich

9. Empirical sociology and structural-functional analysis In the first half of the 20th century, empirical sociology developed rapidly in the West, primarily in Europe and America. It represents a modern manifestation of sociological positivism, the beginning

Functional act

From the book Algorithms of the Mind author Amosov Nikolay Mikhailovich

Functional act Any intellect functions discretely. More precisely, it is a combination of continuous and discrete processes. However, do purely continuous processes exist at all? In any case, in complex systems any continuous is only

2.1.2. Functional analysis

author Isaev Boris Akimovich

2.1.2. Functional Analysis The term “function” means “execution”. In a social system, a function means the fulfillment of roles and certain activities performed by elements in the interests of the system. The essence of functional analysis lies in the identification of elements

2.1.3. Structural-functional analysis

From the book Sociology [ Short course] author Isaev Boris Akimovich

2.1.3. Structural-Functional Analysis We have deliberately presented R. Merton as a proponent of both structural and functional approaches. Indeed, in 1949 he published Paradigms for Functional Analysis and appeared to the world social sciences as sequential

Structural-functional analysis

From the book Great Soviet Encyclopedia (ST) by the author TSB

"Functional analysis and its applications"

TSB

Functional analysis (mathematical)

From the book Great Soviet Encyclopedia (FU) by the author TSB

Functional analysis (chemical)

From the book Great Soviet Encyclopedia (FU) by the author TSB

Functional Act

From the book Encyclopedia of Amosov. Health algorithm author Amosov Nikolay Mikhailovich

Functional Act The mind does not act continuously, but in “portions”, “units of action”. I called them Functional Acts (FA). They are like certain units of thinking mechanisms and require detailed consideration. PA procedures are carried out with the participation of both devices

3.1.3. Functional analysis

From the book Self-Assertion of a Teenager author Kharlamenkova Natalya Evgenevna

3.1.3. Functional analysis Self-affirmation gives rise to a feeling in a person self-esteem and confirms it at different stages of life. Researchers come to this conclusion when they talk about the functions of self-affirmation. If we consider this the only conclusion, then in this