Przestrzeń ekonomiczna, czyli algebra liniowa ciągów finitnych - Smoluk, Antoni (Economical space or linear algebra of finite sequences, 2007)
Vector in economics is a basket of goods; linear space R" is a family of all n-commodity baskets: n-tuples. In an econometrical model the dimension of basket space is usually fixed. It is not convenient because in the flow of time new products appear and other disappear, the number of components of vectors are changeable. In the paper we propose new candidate for the economic space; it is the linear space FF(N, R) of all sequences with terms almost all equal zero; they are really the finite sequences, for example a = (1, 2, 1, 0, 0, ...). Of course linear space R" is imbedded in linear algebra FF(N, R). In this algebra we can introduce three different norms: integral norm ^ |a,. | , Euc- FF(N, R) - three different Banach spaces: L jN , R) - the linear algebra of all summable real sequences, L2(N, R) - the linear space of sequences summable with square and c0(jV, R) - the linear space of sequences which are convergent to zero; of course: L t(N, R) <z L fN , R) c. c0(A, R). The method of completing metric spaces is a useful and powerful instrument in the science. We receive fundamental theorems concerning convergent sequences and series without hard work. The results are important and useful for econometrics and statistics. To end we call the definition of the completing of a metric space: the complete metric space Y is called a completing of the space X if there is an injection f. X —* Y, which is isometry and image of X is dense in Y. Every metric space can be completed.
Prace Naukowe Akademii Ekonomicznej we Wrocławiu; 2007; nr 1189, s. 11-17