On an iterative method for solving the time-optimal problem

The various properties of topological spaces in the classical Baire category theorems which imply the Baire property also imply the stronger property of pseudo-completeness. In contrast to some of these properties and to the Baire property, J. C. Oxtoby has shown that pseudo-completeness is productive. The following main result places pseudo-completeness in the context of linear topological spaces: Let E and F be linear topological spaces and g be a continuous linear mapping of E into F. If E is pseudo-complete, g is almost open, and the completion of g[E] has a continuous metric, then g[E] is complete. The proof of this result uses the difference theorem, but not an open mapping theorem. The hypotheses lead to a discussion of conditions for a linear mapping to be almost open and for a linear topological space to have a continuous metric. An example shows that, although a translation invariant continuous metric on a linear topological space E extends to a translation invariant continuous pseudo-metric on the completion of E, this extension need not be a metric, even if it induces a normed topology on E. Othes examples show that a pseudo-complet e linear topological space need not be complete in its natural uniformity, and that the almost open condition of the main result may not be omitted and is not implied by the combination of the other conditions and the conclusion. Pseudo-complete topological spaces, introduced by Oxtoby [8], successfully bind together the classical Baire category theorems. They have been the subject of recent work; Aarts and Lutzer [1], for example, have shown that a metrizable space is pseudo-comple te if and only if it has a dense subspace which is topologically complete. In the context of linear topological spaces, Todd [13] shows that, under certain conditions, a linear topological space which is pseudocomplete is complete in the natural uniformity. Theorem 3.1 extends this result for the image of a pseudo-complete linear topological space under a continuous linear almost open mapping; in contrast. Example 5.5 gives pseudo-complete locally convex linear topological spaces which are not complete, and hence are not Ptak spaces. From open mapping theorems arise restrictions on continuous images of certain types of spaces. As an application of his open mapping theorem, Banach ([2], p. 38), showed that the continuous linear image of a complete metric linear space in a space of the same

Approximate solution of the time-optimal problem involving delay

The convexity of A and B assures int A+ B=int( A+ B)

In a Hilbert space, we can introduce the notion of orthogonal coordinates through an orthogonal base, and these coordinates are the values of bounded linear functionals defined by the vectors of the base. This suggests that we consider continuous linear functionals, in a linear topological space, as generalized coordinates of the space. To ensure the existence of non-trivial continuous linear functionals in a general locally convex linear topological space, we must rely upon the Hahn-Banach extension theorems.

Nonlinear mappings between linear or linear topological spaces are considered, and the existence of the divided differences is studied.
 For the case of the normed spaces, besides the existence, is also proved an analogical property with the case of the real functions.

Another observation on conditions assuring int A + B = int( A + B)

In this article a local characterization theorem is given for closed sets in a linear topological space that have recession cones with nonempty interior.This theorem is then used to characterize the class of upper semicontinuous increasing functions defined on closed E% -recessional subsets of Ed.1. Introduction.Let S be a closed set in a linear topological space L and let x be a fixed point in S. A vector y is called a direction of recession in S at x if for each positive X the vector x + Xy is in S. If S is convex, then the directions of recession are independent of x and form a closed convex cone called the recession cone of S. The notion of recession cone has been used to sharpen many of the classical results of convexity theory.For example, the following version of Helly's Theorem holds [3]:Let {C; i E 1} be a collection of nonempty closed convex sets in d-dimensional Euclidean space.If the sets have no nontrivial common direction of recession and if each subcollection of at most d + 1 sets has a nonempty intersection, then {C; i E I) has a nonempty intersection.If S is nonconvex, then the directions of recession in S at a fixed point do not necessarily form a convex cone; furthermore, the directions of recession at one point do not determine the directions of recession at different points.However, it is easy to see that the common directions of recession do determine a closed convex cone.When can we conclude that a particular closed convex cone of recession at a fixed point of S is shared by the other points of S ?It is the purpose of this paper to answer this question.Moreover, we use the theory that we develop to characterize an important class of functions in mathematical economics.Before proceeding, we set forth some terminology.If S is a subset of a linear space L, then conv S and cone S will denote the convex hull and the cone generated by S, respectively.If a and b are two vectors in L, then seg[a, b] will denote the line segment joining a and b, whereas seg(a, b) will mean the line segment exclusive of the endpoints.We say that "a sees b via S" whenever seg [a, b] c S. A real valued function/is concave on seg[a, b] if

k-Dimensional intersections of convex sets and convex kernels

The apparatus of relative homological algebra is used to study the derived functors for the functor in the category of linear topological spaces. The cohomology with coefficients in a linear topological space is defined for a linear topological space , and the isomorphism is established. Bibliography: 6 titles.

One of the standard goals of in nite-dimensional topology is the topological classi cation of linear metric spaces and their convex subsets. The origins of this problem go back to the questions of Fr echet [Fr] and Banach [Ba] whether all in nite-dimensional separable Fr echet (Banach) spaces are homeomorphic (we explain our terminology at the end of the introduction). These questions have been a rmatively answered by the combined results of Anderson and Kadec (see [BP]) which may be stated as follows: every separable Fr echet space is homeomorphic to a Hilbert space. This result was generalized by Toru nczyk [To] for the nonseparable case. Since then the main interest in the eld has turned to topological classi cation of incomplete or nonlocally convex linear metric spaces. Quite recently, Cauty [Ca2] has constructed an example of a separable linear complete metric space X which is not an AR. In particular, X is not homeomorphic to any convex subset of a locally convex linear metric space. The topological classi cation of incomplete locally convex metric linear spaces, or even incomplete pre-Hilbert spaces, is much more complex than for the complete ones. First, there exist pre-Hilbert spaces of an arbitrary Borel class. Moreover, the Borel type invariant does not classify such spaces, and in the class of -compact spaces there exist uncountably many pairwise nonhomeomorphic such spaces [BP], [Ca1]. Until recently, there was hope that, in spite of these di culties, the classi cation problem could be restricted to subspaces of Hilbert spaces. The result of Bessaga and Dobrowolski [BD] stating that every locally convex linear -compact metric space is homeomorphic to a pre-Hilbert space looked like a promising rst step. The main objective of this paper is to show that, in general, it is not possible to restrict ones attention to subspaces of Hilbert spaces only. We show that the classes of pre-Hilbert,

Following Kay and Womble [2] an abstract convexity structure on a set X is a collection ξ of subsets of X which includes the empty set, X and is closed under arbitrary intersections. One of the natural problems that arises in convexity structures is to give necessary and sufficient conditions for the existance of a linear structure on X such that the collection of all convex sets in the resulting linear space is precisely ξ. An associated problem is to consider a set with a convexity structure and a topology and find necessary and sufficient conditions for the existance of a linear structure on X such that X becomes a linear topological space with again ξ the collection of convex sets.

Let $E$ and $F$ be locally convex spaces and $G$ their completed $\varepsilon$-tensor product. It is shown that if $S$ and $T$ are weakly almost periodic equicontinuous semigroups of operators on $E$ and $F$ respectively, then, under mild restrictions on $E$ or $F, S \otimes T$ is a weakly almost periodic equicontinuous semigroup of operators on $G$, and the almost periodic and flight vector subspaces of $G$ are related in a natural way to the corresponding subspaces of $E$ and $F$ via the $\varepsilon$-tensor product. Furthermore, if $E$ and $F$ both decompose into a direct sum of these subspaces then so does $G$.

The article introduces the concept of an interior point of a cone of a partially ordered space, which makes it possible to determine the topology in it. This topology turns a given linear space into a topological linear space, which is the main achievement of the article.

Discrete control problems with phase constraints

An iterative numerical solution method is proposed for the general linear quadratic optimal control problem with phase constraints. The method is based on the extension principle. Realization features of the method are described for Matlab environment.