What does functional analysis study? Learning functional, first of all, what does functional study? It can be explained by the following figure: 1. Mapping refers to operators and functionals. 2. Space: X is a collection of some objects defined in a number field. If x is a linear space, then it is a linear space with distance. If we give a norm to X, it is a normed linear space. Assigning an inner product to X is an inner product space (also a norm linear space). Students who control the direction can refer to the textbook "Applied Functional Analysis-Mathematical Basis of Automatic Control". Author: Tsinghua University Publishing House Author: Han (Jiaotong University) This book can be read by graduate students and doctoral students. What is the functional analysis in this paragraph? Functional analysis is a branch of modern mathematics, which belongs to analysis, and its main research object is the space where functions are formed. Functional analysis is developed from studying the properties of transformation (such as Fourier transform) and studying differential equations and integral equations. Using functional as expression comes from variational method, which represents the function used for function. StefanBanach is one of the main founders of functional analysis theory, and mathematician and physicist VitoVolterra has made great contributions to the wide application of functional analysis. From a modern point of view, functional analysis mainly studies complete normed linear spaces in real number fields or complex number fields. This functional analysis space is called Banach space, and the most important special case in Banach space is called Hilbert space, and the norm on it is derived from an inner product. This kind of space is the basis of mathematical description of quantum mechanics. More general functional analysis also studies Fr? Chet space and topological vector space have no defined norm. An important object of functional analysis is continuous linear operators in Banach spaces and Hilbert spaces. This kind of operators can derive the basic concepts of C* algebra and other operator algebras. Hilbert space Hilbert space can be completely classified by the following conclusion: for any two Hilbert spaces, if the cardinality of their bases is equal, then they must be isomorphic. For finite dimensional Hilbert space, the continuous linear operator on it is the linear transformation studied in linear algebra. For infinite dimensional Hilbert space, any morphism on it can be decomposed into countable dimensional (radix 50) morphisms, so functional analysis mainly studies Hilbert space and its countable dimensional morphisms. An unsolved problem in Hilbert space is whether there is a truly invariant subspace for every operator in Hilbert space. The answer to this question in some specific cases is yes. The general Banach space is complex, for example, there is no universal method to construct a set of bases on it. For each real number p, if p? 1, an example of Banach space is the space formed by Lebesgue measurable function, and the integral functional analysis of all its absolute values converges to p power. (See Lp space) In Banach space, a considerable part of research involves the concept of dual space, that is, the space formed by all continuous linear functionals in Banach space. The dual space of dual space may be different from the original space, but the homomorphism of a dual space from Banach space to its dual space can always be constructed. The concept of differential can be generalized in Banach spaces. The differential operator acts on all functions, and the differential of functions at a given point is a continuous linear mapping. The main results and theorems of editing this paragraph include: 1. Uniformly defined theorem (also known as * * * Ming theorem) describes the properties of a family of bounded operators. 2. Spectral theorem includes a series of results, among which the most commonly used result gives the integral expression of normal operators in Hilbert space, which plays a core role in the mathematical description of quantum mechanics. 3.Hahn-BanachTheorem studies how to extend operators from subspace to the whole space in a norm-preserving way. Another related result is the nontrivial nature of dual space. 4. open mapping theorem and closed image theorem. The spaces studied by functional analysis and axiom of choice functional analysis in this paragraph are mostly infinite. In order to prove the existence of a set of bases in infinite dimensional vector space, Zuo En theorem must be used. In addition, most important theorems in functional analysis are based on the Hanbanach theorem, which is itself a form, that is, axiom of choice is weaker than boolean prime ideal theorem. Edit the research status of functional analysis in this paragraph Functional analysis currently includes the following branches: 1. Softanalysis, whose goal is to express mathematical analysis in the language of topological group, topological ring and topological vector space. 2. Jeanbourmain's series of works represent the geometric structure of Banach space. 3. noncommutative geometry, the main contributors in this direction include AlainConnes, and part of his work is based on the results of GeorgeMackey's ergodic theory. 4. Theories related to quantum mechanics are called mathematical physics in a narrow sense. From a broader perspective, as described by IsraelGelfand, they contain most types of problems in the representation theory. Since19th century, the development of mathematics has entered a new stage. That is, due to the study of Euclid's fifth postulate, a new discipline-non-Euclid geometry was introduced. For the general idea of solving algebraic equations, group theory was finally established and developed; The research of mathematical analysis established set theory. These new theories provide conditions for generalizing the basic concepts and methods of classical analysis from a unified point of view. At the beginning of this century, the seeds of generalized analytical science appeared in the works published by Swedish mathematician fleet Holm and French mathematician Adama. Later, Hilbert and Hailingzhe came to study the creation of Hilbert space. In the 1920s, the basic concept of general analysis, namely functional analysis, was gradually formed in the field of mathematics. Due to the formation of many new departments in analysis, many concepts and methods of analysis, algebra and set are often similar. For example, the successive approximation method can be used to find the roots of algebraic equations and solve differential equations, and the conditions for the existence and uniqueness of solutions are very similar. This similarity is more prominent in the theory of integral equation. The emergence of functional analysis is precisely related to this situation, and some seemingly unrelated things have similarities. Therefore, it inspires people to explore universal and truly essential things from these similar things. The establishment of non-Euclidean geometry broadens people's cognition of space, and the appearance of n-dimensional space geometry allows us to interpret multivariate functions as the influence of multi-dimensional space in geometric language. In this way, the similarity between analysis and geometry is shown, and there is the possibility of geometric analysis. This possibility needs to further popularize the concept of geometry, and finally expand Euclidean space into an infinite dimensional space. At this time, the concept of function is given a more general meaning, and the concept of function in classical analysis refers to a corresponding relationship established between two data sets. However, the development of modern mathematics requires the establishment of correspondence between two arbitrary sets. Here we first introduce the concept of operator. Operators are also called operators. Mathematically, the transformation from infinite dimensional space to infinite dimensional space is called operator. Studying the theory of universal functions and operators in infinite dimensional linear space has produced a new analytical mathematics called functional analysis. In 1930s, functional analysis has become an independent subject in mathematics. The feature of functional analysis and content functional analysis in this paragraph is that it not only summarizes the basic concepts and methods of classical analysis, but also geometrizes them. For example, different types of functions can be regarded as points or vectors in the function space, so as to finally get the general concept of abstract space. It includes not only the geometric objects discussed earlier, but also different function spaces. Functional analysis is a powerful tool to study modern physics. N-dimensional space can be used to describe the motion of a mechanical system with n degrees of freedom. In fact, new mathematical tools are needed to describe a mechanical system with infinite degrees of freedom. For example, the vibration of a beam is an example of a mechanical system with infinite degrees of freedom. Generally speaking, the transition from particle mechanics to continuum mechanics requires a transition from a system with finite degrees of freedom to a system with infinite degrees of freedom. The quantum field theory in modern physics belongs to an infinite degree of freedom system. Just as studying a system with limited degrees of freedom needs geometry and calculus in n-dimensional space as tools, studying a system with unlimited degrees of freedom needs geometry and analysis in infinite space, which is the basic content of functional analysis. Therefore, functional analysis can also be popularly called geometry and calculus in infinite dimensional space. The basic method in classical analysis, that is, using linear objects to approximate nonlinear objects, can be fully applied to functional analysis. Functional analysis is the youngest branch of analytical mathematics and a generalization of classical analysis. It integrates the viewpoints of function theory, geometry and algebra to study functions, operators and limit theory in infinite vector space. In the forties and fifties, it became a mathematics subject with complete theory and rich content. For more than half a century, on the one hand, functional analysis has extracted its research objects and some research methods from materials provided by many other disciplines, and formed many important branches of its own, such as operator spectrum theory, Banach algebra, topological linear space theory, generalized function theory, etc. On the other hand, it has also effectively promoted the development of many other analytical disciplines. It has important applications in differential equations, probability theory, function theory, continuum mechanics, quantum physics, computational mathematics, cybernetics, optimization theory and other disciplines. It is also a basic tool for establishing harmonic analysis theory on groups and one of the important and natural tools for studying infinite freedom physical systems. Today, its viewpoints and methods have penetrated into many engineering disciplines and become one of the foundations of modern analysis. Functional analysis is widely used in mathematical physical equations, probability theory, computational mathematics, continuum mechanics, quantum physics and other disciplines. In recent ten years, functional analysis has been more effectively applied in engineering technology. It also permeates all branches of mathematics and plays an important role. |||||| The content of functional analysis For more than half a century, on the one hand, functional analysis has extracted its own research objects and some research methods from materials provided by many other disciplines, and formed many important branches of its own, such as operator spectrum theory, Banach algebra, topological linear space theory, generalized function theory, etc. On the other hand, it has also effectively promoted the development of many other analytical disciplines. It has important applications in differential equations, probability theory, function theory, continuum mechanics, quantum physics, computational mathematics, cybernetics, optimization theory and other disciplines. It is also a basic tool for establishing harmonic analysis theory on groups and one of the important and natural tools for studying infinite freedom physical systems. Today, its viewpoints and methods have penetrated into many engineering disciplines and become one of the foundations of modern analysis. Functional analysis is widely used in mathematical physical equations, probability theory, computational mathematics, continuum mechanics, quantum physics and other disciplines. In recent ten years, functional analysis has been more effectively applied in engineering technology. It also permeates all branches of mathematics and plays an important role. |||||| FunctionalAnalysis is a branch of modern mathematics, which belongs to analysis, and its main research object is the space composed of functions. Functional analysis is developed from studying the properties of transformation (such as Fourier transform) and studying differential equations and integral equations. Using functional as expression comes from variational method, which represents the function used for function.