Fokker-Planck equations. We describe a system evolving as a Markov process on the space of finite configurations, whereas its rates depend on the actual state of another equilibrium process on the space of locally finite configurations. We will show that ergodicity of the environment process implies the averaging principle for the solutions of the coupled Fokker-Planck equations.

Operator representation of Cole-Hopf transform is obtained based on the logarithmic representation of infinitesimal generators.

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For this purpose the relativistic formulation of abstract evolution equation is introduced. Even independent of the spatial dimension, the Cole-Hopf transform is generalized to a transform between linear and nonlinear equations defined in Banach spaces. 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 has a subsequence converging to a point in , 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 , , 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. By definition, ; the topology in , roughly speaking, is given by the convergence which means that with respect to the norm in every.

By definition, ; the topology in , roughly speaking, is given by the convergence which means that all the lie in a certain and that with respect to the norm of this space. 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 with the property that, for each , one can find a such that and the imbedding operator is a Hilbert—Schmidt operator see below, Section.

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. 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.

Let be a Banach space and let be the set of continuous linear functionals on it; 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. The space is called the dual of cf. If is finite-dimensional, then every linear functional is of the form.

It turns out that the formula also holds when is a Hilbert space Riesz' theorem. Namely, in this case , where is a certain vector in. 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 , and these spaces may turn out to be all different. At the same time, there always exists a canonical imbedding of into , namely, to each one can associate the functional , where ,.

The spaces for which 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 is, in a certain sense, "better" than the original space. For example, along with the norm one can introduce another weak topology in which, in terms of convergence, is such that if for all. In this topology the unit ball in 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 the dual space can be found explicitly.

## Functional analysis

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.

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For many questions in functional analysis and its applications an essential role is played by a triple of spaces , where is the original Hilbert space, is a topological vector space in particular, a Hilbert space with a different inner product and is its dual space, the elements of which can be taken as generalized functions. 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. 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.

## Functional Analysis and its Applications

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.

## "Local Fractional Functional Analysis and Its Applications" by Yang Xiao-Jun

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. Harmonic maps from the disk into the Euclidean Nsphere. Nonlinear semigroups and evolution governed by accretive operators. A theorem of Mather and the local structure of nonlinear Fredholm maps. Remarks on Ssup1 symmetries and a special degree for Ssup1invariant gradient mappings. Abstract differential equations maximal regularity and linearization.

Positive solutions for some classes of semilinear elliptic problems. Existence and containment of solutions to parabolic systems. Periodic Hamiltonian trajectories on starshaped manifolds. Integrability of nonlinear differential equations via functional analysis. Extinction of the solutions of some quasilinear elliptic problems of arbitrary order.

### Description

On a system of degenerate diffusion equations. Pointwise continuity for a weak solution of a parabolic obstacle problem. A note on iteration for pseudoparabolic equations of fissured media type. Homogenization of twophase flow equations.