@MISC{EUCLID_7.MIZ,
  AUTHOR = {Nakamura, Yatsuka and Korni{\l}owicz, Artur and Oya, Nagato and Shidama, Yasunari},
  TITLE = {The Real Vector Spaces of Finite Sequences are Finite Dimensional},
  SECTION1 = {Preliminaries},
  SECTION2 = {Orthogonal Basis},
  SECTION3 = {Linear Manifolds},
  SECTION4 = {Standard Basis},
  SECTION5 = {Finite Real Unitary Spaces and Finite Real Linear Spaces},
  SECTION6 = {Finite Dimensionality of the Spaces},
  DAY = {23},
  MONTH = {September},
  YEAR = {2008},
  ADDRESS1 = {Shinshu University\\ Nagano, Japan},
  ADDRESS2 = {Institute of Computer Science\\ University of Bia{\l}ystok\\ Sosnowa 64, 15-887 Bia{\l}ystok, Poland},
  ADDRESS3 = {Shinshu University\\ Nagano, Japan},
  ADDRESS4 = {Shinshu University\\ Nagano, Japan},
  SUMMARY = {In this paper we show the finite dimensionality of real linear spaces 
  with their carriers equal ${\cal R}^n$. We also give the standard basis of such 
  spaces. For the set ${\cal R}^n$ we introduce the concepts of linear manifold 
  subsets and orthogonal subsets. The cardinality of orthonormal basis of 
  discussed spaces is proved to equal~$n$.}}

