Stefan Banach is one of the main founders of functional analysis theory, and Vito Volterra, a mathematician and physicist, has made great contributions to the wide application of functional analysis.
Functional analysis is a branch of mathematics formed in 1930s. It is developed from the study of variational problems, integral equations and theoretical physics. It comprehensively applies the function theory,
Geometry,
Algebraic point of view to study functions in infinite dimensional vector space,
Operator and limit theory. It can be regarded as analytic geometry and mathematical analysis of infinite dimensional vector space. The main contents are topological linear space and so on. Functional analysis has applications in mathematical physics equations, probability theory, computational mathematics and other branches, and it is also a mathematical tool for studying infinite freedom physical systems. Functional analysis is a branch of studying the mapping from topological linear space to topological linear space satisfying various topological and algebraic conditions.
The basis of mathematical description of quantum mechanics. More general functional analysis also studies spaces with undefined norms, such as Frescher space and topological vector space.
An important goal of functional analysis is the sum of Banach spaces.
Hilbert space
Continuous linear operators on. This kind of operators can derive the basic concepts of C* algebra and other operator algebras.
1. Hilbert space
Hilbert spaces can be completely classified by the following conclusion: for any two Hilbert spaces, if the cardinality of their bases is equal, 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.
2. Banach space
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 a space composed of "Lebesgue measurable functions whose absolute values converge to the power integral of p". (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.
Lima, Zuo En. In addition, the most important theorems in functional analysis are based on Hanbanach theorem, which is a form in itself, and axiom of choice is weaker than boolean prime ideal theorem.
Mathematical physics
From a broader perspective, as israel Gelfand said, it contains most types of problems of representation theory.
In Adama's published works, the seeds of generalized analytical science appeared. Subsequently, Hilbert and Hailingzhe initiated the study 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.
Probability theory, computational mathematics, continuum mechanics and quantum physics are widely used. 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.