On spaces with vector structure
Abstract
In [1], using the notion of linear space of translations of the set over the field, n-dimensional Klein spaces over arbitrary field were defined. In [2] the definition of vector structure over the field was given and used to introduce the concept of n-dimensional generalized elementary Klein space.
The aim of present paper is to define (Section 1) and state some of the properties of, so called, spaces with vector structure, without the use of Klein's ideas. In Section 2 it is shown that afiine and Euclidean space are the examples of such spaces. Other examples are the elliptic and projective space. Using the notion of vector structure, in Section 3 the definition of tangent bundle is given and some properties of it are observed, with the aim to introduce (Section 4) the concept of m-dimensional hyperplane in spaces with vector structure.
References
2. B. Szociński, On some generalization of elementary Klein space, Zeszyty Nauk. Politech. Śląsk. Mat.-Fiz. (in print).
3. B. Szociński, Some properties of the projective space, Demonstratio Math. (in print).
Instytut Matematyki, Politechnika Śląska Poland
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