Continuous linear images of pseudo-complete linear topological spaces

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.

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

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

We prove that the algebraic dimension of the linear span of sums of a subseries convergent series in a Hausdorff topological linear space is either finite or equals 2ℵ°. This result is applied to represent every infinite-dimensional metrizable linear spaee of cardinality 2ℵ° as the direct sum of a sequence of dense subspaces with a strange summability property. Moreover, we show that every infinite-dimensional separable metrizable linear space admits an m-quasi-basis which is not a quasi-basis.

: The generalized additive channel was introduced. It is described by an additive noise process with sample functions inducing a measure on a linear topological vector space, and by a constraint which includes dimensionality. The coding capacity of the matched channel was analyzed, with an exact value obtained for the Gaussian channel and an upper bound for a class of nonGaussian channels. Bounds on the coding capacity for the mismatched Gaussian generalized channel were obtained. In this paper, the exact coding capacity of the mismatched Gaussian generalized channel is determined, along with an upper bound for a class of nonGaussian mismatched channels. The set of admissable constraints is also greatly increased over that considered. Although the treatment here is restricted to noise measures induced on a separable Hilbert space, it can readily be seen that the results extended immediately to the class of linear topological vector spaces considered. The results of the present paper are partly based on the Hilbert space results on information capacity given, for the extension of linear topological vector spaces, one would use the corresponding results given. The focus on Hilbert space is useful for application of the results given here to the discrete-time or continuous-time additive channel.

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.

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.

On an iterative method for solving the time-optimal problem

Most of the spaces of random variables are non-Hausdorff linear topological spaces, whereas a number of useful results of the theory of vector topologies require the Hausdorff separation axiom for the underlying spaces. Therefore we review and complete the machinery of non-Hausdorff linear topological spaces and the techniques for their representation with suitable Hausdorff linear topological spaces, and then we work out some examples, which emphasize the ideas involved in a consistent application of the topological point of view. The final part of the paper is devoted to the discussion of the pathology, which arises when we try to treat the special case of almost everywhere convergence in the same framework.

Various forms of boundedness in rectangular fuzzy n-normed linear spaces of type are examined in this work, where ∗ represents an arbitrary t-norm. Some of these highly general notions of boundedness have no counterpart in the traditional topological metrizable linear spaces. We list the properties of these bounded sets and conduct a comparative analysis of the various forms of boundedness. Among these are a number of ideas on the symmetrical characteristics of the objects under study that come from the classical context and are relevant to the issues of this magazine. We define their consequences and provide instances to show how different these notions are from one another.

пУстьт(с) — клАсс РЕгУ льРНых МАтРИц (сmn) с ДЕИстВИтЕльНыМИ ИлИ кОМплЕксНыМИ ЧлЕНАМИ, УДОВлЕтВОРь УЩИх УслОВИУ lim sup $$\mathop {\lim \sup }\limits_{m \to \infty } \mathop \Sigma \limits_{n = 0}^\infty \left| {c_{mn} } \right| = C.$$ .

In this paper we investigate the problem of the Hyers-Ulam stability of the generalized quadratic functional equation $$f(x+y)+f(x-y)=g(x)+g(y),$$ where $f,g$ are functions defined on a group with values in a linear topological space.

In reminiscence of Ptak's open mapping theorem, a topological space satisfying the open mapping theorem is called a B r space.This paper is devoted to the study of sums and products of B r spaces in the category of topological spaces.We prove that, in general, sums and products of even two B r spaces need no longer be B r .On the other hand, for any B r space E the sum E θ E is again a B r space.Moreover, since Cech complete spaces are known to be B r , we ask whether a sum E θ F is B r provided that E is Cech complete and F is B r .It turns out that, at least in the framework of complete regularity, the answer to this question is in the positive if and only if F is a Baire space.Introduction.The notion of B r spaces has first been introduced by T. Husain in the categories of locally convex vector spaces (see [HuJ) and topological groups (see [Hu 2 ]).Originally, it goes back to V. Ptak's open mapping theorem (see [Kό], p. 35 ff).A Hausdorff topological space E is called a B r space if every continuous, nearly open bijection / from E onto any Hausdorff space F is in fact open.For a survey of the classical theory of B r spaces we refer the reader to Kόthe's book [Kό], where the locally convex case is treated.The linear topological case is investigated in the lecture notes [AEK].B r groups are considered by several authors.See for instance [Hu 2 ], [Ba], [Gr], [Su].In a purely topological context, B r spaces have first been investigated by Weston in [We], although the term B r space is not used there.Translated into the B r terminology, Weston proved that every completely metrizable topological space is a B r space.His result has been generalized by Byczkowski and Pol in [BP], who proved that every Cech complete topological space is a B r space.In [No] we have further generalized this result proving that every Hausdorff, semi-regular topological space densely containing some Cech complete sub space is in fact a B r space.We just mention another generalization of Byczkowski and Pols' result into a somewhat different direction by Wilhelm (cf.[Wi]).In the present paper we examine the invariance of the class of B r spaces under topological sums and products.It turns out that, in general, the sum of even two B r spaces need not be a B r space.A counterexample is given in §2.In §1 we obtain a positive result stating that the sum E θ E

SOME SPECIAL METHODS OF HOMEOMORPHISM THEORY IN INFINITE-DIMENSIONAL TOPOLOGY