Functional analysis

The part of modern mathematical analysis in which the basic purpose is to study functions $ y = f ( x) $ for which at least one of the variables $ x $ or $ y $ varies over an infinite-dimensional space. In its most general form such a study falls into three parts: 1) the introduction and study of infinite-dimensional spaces as such; 2) the study of the simplest functions, namely, when $ x $ takes values in an infinite-dimensional space and $ y $ in a one-dimensional space (these are called functionals (cf. Functional), whence the name "functional analysis" ); and 3) the study of general functions of the type indicated — operators (cf. Operator). Linear functions $ X \ni x \mapsto f ( x) = y \in Y $, i.e. linear operators, have been most completely studied. Their theory is essentially a generalization of linear algebra to the infinite-dimensional case. A combination of the approaches of classical analysis and algebra is characteristic for the methods of functional analysis, and this leads to relations between what are at first glance very distant branches of mathematics.

Functional analysis as an independent mathematical discipline started at the turn of the 19th century and was finally established in the 1920's and 1930's, on the one hand under the influence of the study of specific classes of linear operators — integral operators and integral equations connected with them — and on the other hand under the influence of the purely intrinsic development of modern mathematics with its desire to generalize and thus to clarify the true nature of some regular behaviour. Quantum mechanics also had a great influence on the development of functional analysis, since its basic concepts, for example energy, turned out to be linear operators (which physicists at first rather loosely interpreted as infinite-dimensional matrices) on infinite-dimensional spaces.

1. The concept of a space.

Topological vector spaces (cf. Topological vector space) are the most general spaces figuring in functional analysis. These are vector (linear) spaces $ X $ over the field of complex numbers $ \mathbf C $ (or any other field, for example that of the real numbers, $ \mathbf R $) which are simultaneously topological spaces and where the linear structure and the topology are compatible in the sense that the linear operations are continuous in the topology under consideration. In particular, if $ X $ is a metric space, then one has a metric vector space.

A more particular, but very important, situation arises when the concept of the norm $ \| x \| $ (the length) of a vector $ x \in X $ is introduced axiomatically in a vector space $ X $. A vector space with a norm is called a normed space. It is metrizable. A metric $ \rho $ is introduced by the formula: $ \rho ( x, y) = \| x - y \| $. A vector space with a norm is called a Banach space if it is complete with respect to the metric indicated.

In a large number of problems the situation arises where one can introduce an inner product $ ( x, y) $ for any two vectors in the vector space $ X $, such that this product generalizes the usual scalar product in three-dimensional space. A space provided with an inner product is called a pre-Hilbert space; it is a particular case of a normed space. If this space is complete, then it is called a Hilbert space.

Infinite-dimensional spaces are studied in functional analysis, that is, spaces in which there is an infinite set of linearly independent vectors.

From a geometric point of view the simplest spaces are the Hilbert spaces $ X = H $, which have properties that mostly resemble those of finite-dimensional spaces, because it is possible to introduce a concept similar to that of the angle between two vectors by means of the inner product. In particular, two vectors $ x, y \in H $ are said to be orthogonal $ ( x \perp y) $ if $ ( x, y) = 0 $. The following result is true in $ H $: Let $ G $ be a subspace of $ H $, then any vector $ x \in H $ has a projection $ x _ {G} $ onto $ G $, that is, a vector $ x _ {G} \in G $ such that $ x - x _ {G} $ is orthogonal to any vector in $ G $. Due to this fact, a large number of geometric constructions which hold for finite-dimensional spaces can be transferred to Hilbert spaces, where they often acquire an analytic character.

Geometric questions become distinctly more complicated when going from Hilbert spaces to Banach spaces, and all the more so in general topological vector spaces, because orthogonal projection is not meaningful in them. For example, in the space $ l _ {p} $ ($ p \geq 1 $) the vectors $ e _ {n} = ( 0, \dots, 0, 1, 0, 0 , \dots ) $ form a basis in the sense that for each vector $ x \in l _ {p} $ "coordinate-wise" expansion is valid:

$$ x = \ \sum _ {n = 1 } ^ \infty x _ {n} e _ {n} . $$

The construction of a basis for the space $ C [ a, b] $ is already a bit more complicated; at the same time a basis can be constructed in each of the known examples of Banach spaces. The problem arose: Does there exist a basis in every Banach space? This problem, in spite of the efforts of many mathematicians, did not yield a solution for more than 40 years and was only solved negatively in 1972 (see [23]). In functional analysis an important place is occupied by "geometric" themes, devoted to clarifying the properties of various sets in Banach and other spaces, for example convex sets, compact sets (the latter means that every sequence of points of such a set $ Q $ has a subsequence converging to a point in $ Q $), etc. Here, simply formulated questions often have very non-trivial solutions. These problems are closely connected with the study of isomorphisms between spaces, and with finding universal representatives in some classes of spaces.

Specific function spaces have been studied in detail, since the properties of these spaces usually determine the character of the solution to a problem when it is obtained by the methods of functional analysis. The so-called imbedding theorems for the Sobolev spaces $ W _ {p} ^ {l} ( G) $, $ G \subseteq \mathbf R ^ {n} $, and various generalizations of these, can serve as an example.

In connection with the demands of modern mathematical physics a great number of specific spaces have arisen in which problems are naturally posed and which thus must be studied. These spaces are usually constructed from initial spaces using certain constructions. Below the most commonly used constructions are given in their simplest versions.

1) The formation of an orthogonal sum $ H = \oplus _ {n = 1 } ^ \infty H _ {n} $ of Hilbert spaces $ H _ {n} $, $ n = 1, 2, \dots $ is a construction of a space $ H $ in terms of spaces $ H _ {n} $, similar to the formation of $ H $ in terms of one-dimensional spaces.

2) Passing to a quotient space: Given a degenerate inner product $ ( x, y) $ in a vector space $ X $ (that is, $ ( x, x) = 0 $ is possible when $ x \neq 0 $); the Hilbert space $ H $ is defined as the completion of $ X $ with respect to $ ( \cdot , \cdot ) $ after first identifying with 0 all those vectors for which $ ( x, x) = 0 $.

3) The formation of a tensor product $ \otimes _ {j = 1 } ^ {n} H _ {j} $ is analogous to passing from functions of one variable $ f ( x _ {1} ) $ to functions of several variables $ f ( x _ {1}, \dots, x _ {n} ) $; a similar construction is also used for an infinite number of factors; one also considers symmetric and anti-symmetric tensor products consisting, in the case of functions, of functions of several variables having these properties.

4) The formation of the projective limit $ X $ of Banach spaces $ X _ \alpha $, where $ \alpha $ runs over a certain set of indices $ A $. By definition, $ X = \cap _ {\alpha \in A } X _ \alpha $; the topology in $ X $, roughly speaking, is given by the convergence $ x _ {n} \rightarrow x $ which means that $ \| x _ {n} - x \| \rightarrow 0 $ with respect to the norm in every $ X _ \alpha $.

5) The formation of the inductive limit $ X $ of the Banach spaces $ X _ \alpha $. By definition, $ X = \cup _ {\alpha \in A } X _ \alpha $; the topology in $ X $, roughly speaking, is given by the convergence $ x _ {n} \rightarrow x $ which means that all the $ x _ {n} $ lie in a certain $ X _ \alpha $ and that $ \| x _ {n} - x \| \rightarrow 0 $ with respect to the norm of this space.

6) Interpolation is the formation of "intermediate" spaces $ X _ \alpha $ from two spaces $ X _ {1} $ and $ X _ {2} $, where $ \alpha \in ( 1, 2) $; for example, the construction from $ W _ {p} ^ {1} ( G) $ and $ W _ {p} ^ {2} ( G) $ of the space $ W _ {p} ^ \alpha ( G) $ of functions with fractional derivative $ \alpha \in ( 1, 2) $.

Procedures 4) and 5) are commonly applied when constructing topological vector spaces. One distinguishes among such spaces the very important class of the so-called nuclear spaces (cf. Nuclear space), each of which is constructed as a projective limit of Hilbert spaces $ H _ \alpha $ with the property that, for each $ \alpha \in A $, one can find a $ \beta \in A $ such that $ H _ \beta \subseteq H _ \alpha $ and the imbedding operator $ H _ \beta \ni x \rightarrow x \in H _ \alpha $ is a Hilbert–Schmidt operator (see below, Section $ \mathbf 3 $).

An extensive and important branch of functional analysis has been developed in which one studies topological and normed vector spaces with a partial order, introduced axiomatically, having natural properties (partially ordered spaces).

2. Functionals.

In functional analysis the study of continuous functionals and linear functionals plays an essential role (cf. Continuous functional; Linear functional); their properties are closely connected with the properties of the original space $ X $.

Let $ X $ be a Banach space and let $ X ^ \prime $ be the set of continuous linear functionals on it; $ X ^ \prime $ is a vector space with respect to the usual operations of adding functions and multiplying them by a number, it becomes a Banach space if one introduces the norm

$$ \| l \| = \ \sup _ {\begin{array}{c} x \in X \\ x \neq 0 \end{array} } \ \frac{| l ( x) | }{\| x \| } . $$

The space $ X ^ \prime $ is called the dual of $ X $ (cf. also Adjoint space).

If $ X $ is finite-dimensional, then every linear functional is of the form

$$ l ( x) = \ \sum _ {j = 1 } ^ { n } x _ {j} \overline{a} _ {j} , $$

where $ x _ {j} $ are the coordinates of the vector $ x $ with respect to a certain basis and $ a _ {j} $ are numbers determined by the functional. It turns out that the formula also holds when $ X = H $ is a Hilbert space (Riesz' theorem). Namely, in this case $ l ( x) = ( x, a) $, where $ a $ is a certain vector in $ H $. This formula shows that a Hilbert space essentially coincides with its dual.

For a Banach space the situation is far more complicated: One can construct $ X ^ {\prime\prime} = ( X ^ \prime ) ^ \prime $, $ X ^ {\prime\prime \prime } = ( X ^ {\prime\prime} ) ^ \prime , \dots $ and these spaces may turn out to be all different. At the same time, there always exists a canonical imbedding of $ X $ into $ X ^ {\prime\prime} $, namely, to each $ x \in X $ one can associate the functional $ L _ {x} $, where $ L _ {x} ( l) = l ( x) $, $ l \in X ^ \prime $. The spaces $ X $ for which $ X ^ {\prime\prime} = X $ are called reflexive. Generally, in the case of a Banach space even the existence of non-trivial (that is, non-zero) linear functionals is not a simple question. This question is easily solved affirmatively with the help of the Hahn–Banach theorem.

The dual space $ X ^ \prime $ is, in a certain sense, "better" than the original space $ X $. For example, along with the norm one can introduce another (weak) topology in $ X ^ \prime $ which, in terms of convergence, is such that $ l _ {n} \rightarrow l $ if $ l _ {n} ( x) \rightarrow l ( x) $ for all $ x \in X $. In this topology the unit ball in $ X ^ \prime $ is compact (which is never the case for infinite-dimensional spaces in the topology generated by a norm). This makes it possible to study in more detail a number of geometric questions about sets in the dual space (for example, establishing the structure of convex sets, etc.).

For a number of specific spaces $ X $ the dual space $ X ^ \prime $ can be found explicitly. However, for the majority of Banach spaces, and especially for topological vector spaces, the functionals are elements of a new kind which cannot be expressed simply in terms of classical analysis. The elements of the dual space are called generalized functions.

For many questions in functional analysis and its applications an essential role is played by a triple of spaces $ \Phi ^ \prime \supseteq H \supseteq \Phi $, where $ H $ is the original Hilbert space, $ \Phi $ is a topological vector space (in particular, a Hilbert space with a different inner product) and $ \Phi ^ \prime $ is its dual space, the elements of which can be taken as generalized functions. The space $ H $ itself is then called a rigged Hilbert space.

The study of linear functionals on $ X $ in many respects promotes a deeper understanding of the nature of the original space $ X $. On the other hand, in many questions it is necessary to study general functions $ X \ni x \rightarrow f ( x) \in \mathbf C $, that is, non-linear functionals in the case of an infinite-dimensional $ X $ (cf. Non-linear functional). Since the unit ball in such a space $ X $ is non-compact, its study often encounters essential difficulties, although, for example, such concepts as the differentiability of $ f $, its analyticity, etc. are easily generalized. One can consider a set of functions $ X \ni x \mapsto f ( x) \in \mathbf C $ having definite properties as a new topological vector space of functions of "an infinite number of variables" . Such functions also appear in constructing infinite tensor products $ \otimes _ {n = 1 } ^ \infty H _ {n} $ of spaces of functions of one variable. The study of such spaces, of the operators on them, etc., is connected with the requirements of quantum field theory (see [22]).

3. Operators.

The main objects of study in functional analysis are operators $ X \ni x \mapsto f ( x) \in Y $, where $ X $ and $ Y $ are topological vector (for the most part, normed or Hilbert) spaces and, above all, linear operators (cf. Linear operator).

When $ X $ and $ Y $ are finite-dimensional, the linearity of an operator implies that it is of the form

$$ ( Ax) _ {j} = \ \sum _ {k = 1 } ^ { N } a _ {jk} x _ {k} ,\ \ j = 1, \dots, M, $$

where $ x _ {1}, \dots, x _ {N} $ are the coordinates of the vector $ x $ in a certain basis, and $ ( Ax) _ {1}, \dots, ( Ax) _ {M} $ are, analogously, the coordinates of $ y = Ax $. Thus, in the finite-dimensional case to each linear operator $ A $ corresponds, in terms of fixed bases in $ X $ and $ Y $, a matrix

$$ \| a _ {jk} \| _ {j, k = 1 } ^ {M, N } , $$

which gives a simple expression for $ A $. The study of linear operators in this case is a topic of linear algebra.

The situation becomes much more complicated when $ X $ and $ Y $ become infinite-dimensional (even Hilbert) spaces. First of all, two classes of operators arise here: continuous operators, for which the function $ X \ni x \mapsto Ax \in Y $ is continuous (they are also called bounded, since the continuity of an operator between Banach spaces is equivalent to its boundedness), and unbounded operators, where there is no such continuity. The operators of the first type are simpler, but those of the second type are met more often, e.g. differential operators are of the second type.

The important (especially for quantum mechanics) class of self-adjoint operators on a Hilbert space $ H $ has been studied most of all (cf. Self-adjoint operator).

Other classes of operators on $ H $, closely connected with the self-adjoint operators (the so-called unitary and normal operators, cf. Unitary operator; Normal operator), have also been well studied.

Among the general facts about bounded operators acting in a Banach space $ X $, one can select the construction of a functional calculus of analytic functions. Namely, the operator $ R _ {z} = ( A - zI) ^ {- 1} $ is called the resolvent of the operator $ A $, where $ I $ is the identity operator and $ z \in \mathbf C $. The points $ z $ for which the inverse operator $ ( A - zI) ^ {- 1} $ exists are called the regular points of $ A $, the complement of the set of regular points is called the spectrum $ s ( A) $ of $ A $. The spectrum is never empty and lies in the disc $ | z | \leq \| A \| $; the eigen values of $ A $, of course, belong to $ s ( A) $, but the spectrum, generally speaking, does not entirely consist of them. If $ f ( z) $ is an analytic function defined in a neighbourhood of $ s ( A) $, and if $ \Gamma $ is some closed contour enclosing $ s ( A) $ and lying in the domain of analyticity of $ f ( z) $, then one puts

$$ f ( A) = \ - { \frac{1}{2 \pi i } } \oint _ \Gamma f ( z) R _ {z} dz $$

and calls $ f ( A) $ an operator function. If $ f ( z) $ is a polynomial, then $ f ( A) $ is obtained by simply replacing $ z $ in this polynomial by $ A $. The correspondence $ f ( z) \mapsto f ( A) $ has the important homomorphism properties:

$$ ( f + g) ( A) = \ f ( A) + g ( A),\ \ ( fg) ( A) = \ f ( A) g ( A). $$

Thus, under definite conditions on $ A $ one can define, for example, $ e ^ {A} $, $ \sin A $, $ \sqrt A $, etc.

Among the special classes of operators acting on a Banach space $ X $ the most important role is played by the so-called completely-continuous or compact operators (cf. Completely-continuous operator; Compact operator). If $ A $ is compact, then the equation $ x - Ax = y $ ($ y \in X $ is a given vector and $ x \in X $ is the desired vector) has been well studied. The analogues of all the facts which hold for linear equations in finite-dimensional spaces are also valid for this equation (the so-called Fredholm theory). For compact operators $ A $ one studies conditions which ensure that the system of eigen vectors of $ A $ and their associated vectors are dense in $ X $, that is, any vector in $ X $ can be approximated by linear combinations of eigen vectors and associated vectors; etc. At the same time there are, even for compact operators, problems which naturally arise but which are very difficult to solve (for example, the theorem that each such operator has an invariant subspace $ G $ different from 0 and the whole of $ X $, that is, a subspace $ G $ such that $ AG \subseteq G $; in the finite-dimensional case the existence of $ G $ follows trivially from the fact that the spectrum is non-empty).

The spectrum of a compact operator $ A $ is discrete and may accumulate at 0 only. One distinguishes important subclasses of the class of compact operators according to the rate at which the eigen values approach 0. Thus, very often one encounters Hilbert–Schmidt operators. If $ A $ is an operator on $ H = L _ {2} ( G) $, then it is a Hilbert–Schmidt operator if and only if it is an integral operator with kernel $ K ( t, s) $ that is square-summable in both variables. Compact Volterra operators have also been studied in detail. A study has also been made of spectral operators for which there is an analogue for the resolution of the identity $ E ( \lambda ) $; etc. (see [8]).

4. Banach algebras and representation theory.

In the early stages of the development of functional analysis the problems studied were those that could be stated and solved in terms of linear operations on elements of the space alone.

One of the powerful methods in mathematics is to represent abstract mathematical objects by simpler (or more concrete) objects. For example, the spectral theorem can be interpretated as representing a self-adjoint operator by the operator which multiplies the measurable functions of a certain class by the independent variable. If one considers multiplication by Borel functions, one obtains a representation of a commutative normed algebra of operators on a Hilbert space. A more general example of this representation gives one of the main theorems in the theory of commutative Banach algebras.

Let $ A $ be a commutative Banach algebra, for simplicity with an identity, that is, a Banach space in which there is a commutative and associative multiplication $ x \cdot y $ of elements $ x, y \in A $, and let the norm satisfy $ \| xy \| \leq \| x \| \cdot \| y \| $. Further, let $ \mathfrak M $ be the set of all maximal ideals. Then a compact topology can be introduced on $ \mathfrak M $ so that every element $ x \in A $ represents a complex-valued continuous function $ x ( m) $, $ m \in \mathfrak M $, and, moreover, the sum $ x ( m) + y ( m) $ and the product $ x ( m) \cdot y ( m) $ of functions correspond to the sum $ x + y $ and the product $ x \cdot y $, respectively (see [7]). In the non-commutative case representation theory has been studied especially for the so-called algebras with an involution (see Banach algebra).

A considerably richer representation theory has been developed for topological groups (cf. Representation of a topological group).

5. Non-linear functional analysis.

At the same time as the concept of a space was being developed and deepened, the concept of a function was being developed and generalized. In the end it became necessary to consider mappings (not necessarily linear) from one space into another. One of the central problems in non-linear functional analysis is the study of such mappings. As in the linear case, a mapping of a space into (the real or complex) numbers is called a functional. For non-linear mappings (in particular, non-linear functionals) there are various methods to define the concepts of a differential, a directional derivative, etc., analogous to the corresponding concepts in classical mathematical analysis (see [11], [13], [15] and Differentiation of a mapping).

An important problem in non-linear functional analysis is the problem of determining the fixed points of a mapping (see [11], [13], [15] and Fixed point).

When studying the eigen vectors of a non-linear mapping containing a parameter there arises a phenomenon that is crucial in non-linear analysis — so-called bifurcation (see [15]).

In the investigation of fixed points and bifurcation points, topological methods are extensively used: the generalization to infinite-dimensional spaces of the Brouwer–Bohl theorem on the existence of fixed points for mappings of finite-dimensional spaces, the index of a mapping (cf. Index formulas), etc.

6. The application of functional analysis in mathematical and theoretical physics.

Below, those branches of mathematical physics are given in which some part of functional analysis is applied.

1) The spectral theory of operators is applied in all theories of quantum physics: in quantum $ n $-body theory, in quantum field theory and in quantum statistical mechanics. In addition, spectral theory is applied in the study of models of dynamical systems in classical mechanics, in the study of linearized equations in hydrodynamics, in the study of Gibbs fields, etc.

2) Scattering theory is applied in quantum physics. It should be noted that the modern mathematical theory of scattering arose first of all in physics. In recent years scattering theory (the inverse problem) has often been applied in integrating non-linear model equations in mathematical physics.

3) Banach algebras are applied in quantum field theory, especially in so-called axiomatic field theory, and in studying various integrable models of a quantum field and statistical mechanics. Von Neumann algebras are also used in these questions.

4) Perturbation theory, mainly the perturbation theory for linear operators, is applied in almost-all domains of mathematical physics: in quantum field theory and in statistical mechanics, both equilibrium and non-equilibrium (especially in investigating the so-called kinetic equations, the compound spectra of multi-particle systems, etc.).

5) Functional integration and measures in function spaces are applied in constructive quantum field theory and in quantum statistical mechanics.

6) Various integral representations (Riesz' theorem, the Krein–Mil'man theorem, Choquet's theorem, and others) are applied in axiomatic quantum field theory and in statistical mechanics.

7) Vector spaces (mainly Hilbert spaces) are applied in quantum theory and in statistical physics.

8) Generalized functions are applied everywhere in mathematical physics as an important analytic tool (see also Generalized function).

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