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Schreier varieties of linear algebras , Soviet Mathematics - Sbornik, , v. S-bounded products of linear W -algebras , Matematicheskiye Issledovaniya, , v. The cancellation law and accessible classes of linear W -algebras , Matematicheskiye Zametki, , v. Non-classical models of natural numbers , Russian Mathematical Surveys, , v. Free sums of principal objects , Notices of the American Mathematical Society, , v.

Compactly generated algebras with continuous operations , International Topological Conference, Moscow , , p.

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Free algebras with continuous systems of operations , Russian Mathematical Surveys, , v. Reduction of algebras with infinite sets of operations , Abstracts of papers presented to the American Mathematical Society, , v.


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Some toirsions in abelian categories with principal objects , Abstracts of papers presented to the American Mathematical Society, , v. Dual arithmetics , Abstracts of papers presented to the American Mathematical Society, , v. Kurosh varieties of linear W -algebras, in "Problems of group theory and homological algebra", Yaroslavl , , pp.

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Some generalizations of the theorems of Cauchy and d'Alembert , Abstracts of papers presented to the American Mathematical Society, , v. Generalized Freiheitssatz in linear W - algebras with graded presentation , Abstracts of papers presented to the American Mathematical Society, , v. Residual properties of linear W -algebras , Abstracts of papers presented to the American Mathematical Society, , v. Schreier and Kurosh varieties of Lie triple systems , Abstracts of papers presented to the American Mathematical Society, , v.

Decompositions of principally q -generated objects , Abstracts of papers presented to the American Mathematical Society, , v. Named sets and their categories , Abstracts of papers presented to the American Mathematical Society, , v. Multicardinals , Abstracts of papers presented to the American Mathematical Society, , v. Principally generated radicals in abelian categories , in "Problems of group theory and homological algebra," Yaroslavl , , pp. New aspects of multicombinatorics , Abstracts of papers presented to the American Mathematical Society, , v.

Functional bisequences , Abstracts of papers presented to the American Mathematical Society, , v. Order cardinality of named sets , Abstracts of papers presented to the American Mathematical Society, , v. Constructive Logic , in "Philosophical Dictionary," Kiev , , pp. Continuum, in "Philosophical Dictionary," Kiev , , p. Algebras of d-distributions , Abstracts of papers presented to the American Mathematical Society, , v.

Universality of algebras of distributions , Abstracts of papers presented to the American Mathematical Society, , v. Upper multicardinals , Abstracts of papers presented to the American Mathematical Society, , v. Logical deductive varieties , Abstracts of papers presented to the American Mathematical Society, , v. Logical model varieties , Abstracts of papers presented to the American Mathematical Society, , v. Induced generalized measures , Abstracts of papers presented to the American Mathematical Society, , v. Functor semantics in categories of named sets, in "Rationality, Reasoning, Communication," Kiev , , pp.

Categories as systems of named sets , Abstracts of papers presented to the American Mathematical Society, , v. Functors as mappings of named sets , Abstracts of papers presented to the American Mathematical Society, , v. Counting operations with named sets , Abstracts of papers presented to the American Mathematical Society, , v. First order model theories , Abstracts of papers presented to the American Mathematical Society, , v. Asymptotic series and w -numbers , Abstracts of papers presented to the American Mathematical Society, , v. Open Preview See a Problem? Details if other :. Thanks for telling us about the problem.

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The space itself is then called a rigged Hilbert space. The study of linear functionals on in many respects promotes a deeper understanding of the nature of the original space. On the other hand, in many questions it is necessary to study general functions , that is, non-linear functionals in the case of an infinite-dimensional cf. Non-linear functional. Since the unit ball in such a space is non-compact, its study often encounters essential difficulties, although, for example, such concepts as the differentiability of , its analyticity, etc.

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One can consider a set of functions 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 of spaces of functions of one variable. The study of such spaces, of the operators on them, etc. The main objects of study in functional analysis are operators , where and are topological vector for the most part, normed or Hilbert spaces and, above all, linear operators cf.

Linear operator. When and are finite-dimensional, the linearity of an operator implies that it is of the form. Thus, in the finite-dimensional case to each linear operator corresponds, in terms of fixed bases in and , a matrix.

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The study of linear operators in this case is a topic of linear algebra. The situation becomes much more complicated when and become infinite-dimensional even Hilbert spaces. First of all, two classes of operators arise here: continuous operators, for which the function 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. The important especially for quantum mechanics class of self-adjoint operators on a Hilbert space has been studied most of all cf. Self-adjoint operator. Other classes of operators on , 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 , one can select the construction of a functional calculus of analytic functions. Namely, the operator is called the resolvent of the operator , where is the identity operator and.

The points for which the inverse operator exists are called the regular points of , the complement of the set of regular points is called the spectrum of. The spectrum is never empty and lies in the disc ; the eigen values of , of course, belong to , but the spectrum, generally speaking, does not entirely consist of them. If is an analytic function defined in a neighbourhood of , and if is some closed contour enclosing and lying in the domain of analyticity of , then one puts. If is a polynomial, then is obtained by simply replacing in this polynomial by.

The correspondence has the important homomorphism properties:. Thus, under definite conditions on one can define, for example, , , , etc. Among the special classes of operators acting on a Banach space the most important role is played by the so-called completely-continuous or compact operators cf.

Completely-continuous operator ; Compact operator. If is compact, then the equation is a given vector and 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 one studies conditions which ensure that the system of eigen vectors of and their associated vectors are dense in , that is, any vector in 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 different from 0 and the whole of , that is, a subspace such that ; in the finite-dimensional case the existence of follows trivially from the fact that the spectrum is non-empty. The spectrum of a compact operator 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 is an operator on , then it is a Hilbert—Schmidt operator if and only if it is an integral operator with kernel 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 ; etc. 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.

Mathematics

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 be a commutative Banach algebra, for simplicity with an identity, that is, a Banach space in which there is a commutative and associative multiplication of elements , and let the norm satisfy.

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Further, let be the set of all maximal ideals. Then a compact topology can be introduced on so that every element represents a complex-valued continuous function , , and, moreover, the sum and the product of functions correspond to the sum and the product , 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. 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.