Let V be a vector space over R. A norm on V is a function || · || : V → R satisfying three properties: 1) ||v|| ≥ 0, with equality if and only if v = 0, 2) ||v + w|| ≤ ||v|| + ||w|| for v, w ∈ V , 3) ||αv|| = |α|||v|| for α ∈ R, v ∈ V. The same definition applies to a complex vector space. From a norm we get a metric on V by d(v, w) = ||v − w||.

Supervised and unsupervised prototype based vector quantization frequently are proceeded in the Euclidean space. In the last years, also non-standard metrics became popular. For classification by support vector machines, Hilbert or Banach space representations are very successful based on so-called kernel metrics. In this paper we give the mathematical justification that gradient based learning...

Abstract. A new supervised adaptive metric approach is introduced for mapping an input vector space to a plottable low-dimensional subspace in which the pairwise distances are in maximum correlation with distances of the associated target space. The formalism of multivariate subspace regression (MSR) is based on cost function optimization, and it allows assessing the relevance of input vector a...

in this paper, we introduce the cone normed spaces and cone bounded linear mappings. among other things, we prove the baire category theorem and the banach--steinhaus theorem in cone normed spaces.

In 1977, M. Matsumoto and R. Miron [9] constructed an orthonormal frame for an n-dimensional Finsler space, called ‘Miron frame’. The present authors [1, 2, 3, 10, 11] discussed four-dimensional Finsler spaces equipped with such frame. M. Matsumoto [7, 8] proved that in a three-dimensional Berwald space, all the main scalars are h-covariant constants and the h-connection vector vanishes. He als...

in this paper, the notion of $psi -$weak contraction cite{rhoades} isextended to fuzzy metric spaces. the existence of common fixed points fortwo mappings is established where one mapping is $psi -$weak contractionwith respect to another mapping on a fuzzy metric space. our resultgeneralizes a result of gregori and sapena cite{gregori}.

‎In this paper, generalized convex contractions on orthogonal metric spaces are stablished in whath  might be called their  definitive versions. Also, we show that there are examples which show that our main theorems are  genuine generalizations of Theorem 3.1 and 3.2 of [M.A. Miandaragh, M. Postolache and S. Rezapour,  {it Approximate fixed points of generalized convex contractions}, Fixed Poi...

Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this article we show how the geometric algebra of a ...

In this paper, we formalize the Menger probabilistic normed space as a category in which its objects are the Menger probabilistic normed spaces and its morphisms are fuzzy continuous operators. Then, we show that the category of probabilistic normed spaces is isomorphicly a subcategory of the category of topological vector spaces. So, we can easily apply the results of topological vector spaces...