Abstract. The aim of this paper is to study the notion of lacunary I-convergence in probabilistic normed spaces as a variant of the notion of ideal convergence. Also lacunary I-limit points and lacunary I-cluster points have been defined and the relation between them has been established. Furthermore, lacunary Cauchy and lacunary I-Cauchy sequences are introduced and studied. Finally, we provid...

The claim that follows, which I have called the nite-dimensional normed linear space theorem, essentially says that all such spaces are topologically R with the Euclidean norm. This means that in many cases the intuition we obtain in R,R, and R by imagining intervals, circles, and spheres, respectively, will carry over into not only higher dimension R but also any vector space that has nite dim...

We prove several results of the following type: given finite dimensional normed space V possessing certain geometric property there exists another space X having the same property and such that (1) log dimX = O(log dimV ) and (2) every subspace of X, whose dimension is not “too small,” contains a further wellcomplemented subspace nearly isometric to V . This sheds new light on the structure of ...

Let us consider two nonempty subsets A, B of a normed linear space X , and let us denote by 2B the set of all subsets of B. We introduce a new class of multivalued mappings {T : A → 2B}, called R-KKM mappings, which extends the notion of KKM mappings. First, we discuss some sufficient conditions for which the set ∩{T (x) : x ∈ A} is nonempty. Using this nonempty intersection theorem, we attempt...

We investigate a new notion of embedding of subsets of {−1, 1}n in a given normed space, in a way which preserves the structure of the given set as a class of functions on {1, ..., n}. This notion is an extension of the margin parameter often used in Nonparametric Statistics. Our main result is that even when considering “small” subsets of {−1, 1}n, the vast majority of such sets do not embed i...

We define embedding of an n-dimensional normed space into L−p, 0 < p < n by extending analytically with respect to p the corresponding property of the classical Lp-spaces. The well-known connection between embeddings into Lp and positive definite functions is extended to the case of negative p by showing that a normed space embeds in L−p if and only if ‖x‖ is a positive definite distribution. U...