AN ANALOGY OF HAHN–BANACH SEPARATION THEOREM FOR NEARLY TOPOLOGICAL LINEAR SPACES

In this paper, we introduce the notion of nearly topological linear spaces and use it to formulate an alternative definition of the Hahn–Banach separation theorem. We also give an example of a topological linear space to which the result is not valid. It is shown that \(\mathbb{R}\) with its ordinary topology is not a nearly topological linear space.

Tilings of Normed Spaces

By a tiling of a topological linear space X, we mean a covering of X by at least two closed convex sets, called tiles, whose nonempty interiors are pairwise disjoint. Study of tilings of infinite dimensional spaceswas initiated in the 1980's with pioneer papers by V. Klee. We prove some general properties of tilings of locally convex spaces, and then apply these results to study the existence of tilings of normed and Banach spaces by tiles possessing certain smoothness or rotundity properties. For a Banach space X, our main results are the following. (i) X admits no tiling by Fréchet smooth bounded tiles. (ii) If X is locally uniformly rotund (LUR), it does not admit any tiling by balls. (iii) On the other hand, some spaces, г uncountable, do admit a tiling by pairwise disjoint LUR bounded tiles.

Multipliers and operators on the tempered ultradistribution spaces of Roumieu type for the distributional Hankel-type transformation spaces

The tempered ultradistribution space of Roumieu type for the spaceHμ,νis defined, which is a subspace of the Hausdörff locally convex topological linear space. Further, results are obtained for the multipliers and operators on the tempered ultradistribution spaces for the distributional Hankel-type transformation spaces.

A Tauberian theorem for statistical convergence

The notion of statistical convergence of a sequence (xk) in a locally convex Hausdorff topological linear space X was introduced recently by Maddox[5], where it was shown that the slow oscillation of (sk) was a Tauberian condition for the statistical convergence of (sk).

Matrix transformations and convex spaces

In a Hausdorff topological linear space we examine relations betweenr-convexity and a condition on matrix transformations between null sequences. In particular, for metrizable spaces the condition impliesr-convexity. For locally bounded spaces the condition implies sequential completeness.

A duality theorem for nondifferentiable convex programming with operatorial constraints

A duality theorem of Wolfe for non-linear differentiable programming is now extended to minimization of a non-differentiable, convex, objective function defined on a general locally convex topological linear space with a non-differentiable operatorial constraint, which is regularly subdifferentiable. The gradients are replaced by subgradients. This extended duality theorem is then applied to a programming problem where the objective function is the sum of a positively homogeneous, lower semi continuous, convex function and a subdifferentiable, convex function. We obtain another duality theorem which generalizes a result of Schechter.

The graph-theoretic analogue of Tietze's characterization of a convex set and its generalization

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)

Convex series

1. Introduction. Let A be a subset of a Hausdorff topological linear space. By a convex series of elements of A we mean a series of the form where an∈A and λn ≥ 0 for each n, and . We say that A is:(i) CS-closed if it contains the sum of every convergent convex series of its elements;(ii) CS-compact if every convex series of its elements converges to a point of A (this bold terminology is chosen because sets satisfying this condition turn out to have properties analogous to those of compact sets).

C(X) As A Dual Space

It is known [1] that for compact Hausdorff X, C(X) is the dual of a Banach space if and only if X is hyperstonian, that is the closure of an open set in X is again open and the carriers of normal measures in C(X)* have dense union in X. With the desiratum of proving that C(X) is always the dual of some sort of space we broaden the concept of Banach space as follows. A Banach space may be comfortably regarded as a pair (E, B) where E is a topological linear space and B is a subset of E ; the requisite property is that the Minkowski functional of B be a complete norm whose topology coincides with that of E.