The Real Vector Spaces of Finite Sequences are Finite Dimensional

Open access

The Real Vector Spaces of Finite Sequences are Finite Dimensional

In this paper we show the finite dimensionality of real linear spaces with their carriers equal Rn. We also give the standard basis of such spaces. For the set Rn we introduce the concepts of linear manifold subsets and orthogonal subsets. The cardinality of orthonormal basis of discussed spaces is proved to equal n.

MML identifier: EUCLID 7, version: 7.11.01 4.117.1046

Keywords:
References
  • [1] Kanchun and Yatsuka Nakamura. The inner product of finite sequences and of points of n-dimensional topological space. Formalized Mathematics, 11(2):179-183, 2003.

  • [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.

  • [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

  • [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

  • [5] Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245-254, 1990.

  • [6] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.

  • [7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.

  • [8] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

  • [9] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

  • [10] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

  • [11] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.

  • [12] Jing-Chao Chen. The Steinitz theorem and the dimension of a real linear space. Formalized Mathematics, 6(3):411-415, 1997.

  • [13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.

  • [14] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.

  • [15] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23-28, 2003.

  • [16] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Linear combinations in real unitary space. Formalized Mathematics, 11(1):17-22, 2003.

  • [17] Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.

  • [18] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.

  • [19] Yatsuka Nakamura. Sorting operators for finite sequences. Formalized Mathematics, 12(1):1-4, 2004.

  • [20] Henryk Oryszczyszyn and Krzysztof Prażmowski. Real functions spaces. Formalized Mathematics, 1(3):555-561, 1990.

  • [21] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.

  • [22] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.

  • [23] Nobuyuki Tamura and Yatsuka Nakamura. Determinant and inverse of matrices of real elements. Formalized Mathematics, 15(3):127-136, 2007, doi:10.2478/v10037-007-00014-7.

  • [24] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.

  • [25] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.

  • [26] Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847-850, 1990.

  • [27] Wojciech A. Trybulec. Binary operations on finite sequences. Formalized Mathematics, 1(5):979-981, 1990.

  • [28] Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581-588, 1990.

  • [29] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297-301, 1990.

  • [30] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.

  • [31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

  • [32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.

  • [33] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

  • [34] Hiroshi Yamazaki, Yoshinori Fujisawa, and Yatsuka Nakamura. On replace function and swap function for finite sequences. Formalized Mathematics, 9(3):471-474, 2001.

Formalized Mathematics

(a computer assisted approach)

Journal Information


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Target Group

researchers in the fields of formal methods and computer-checked mathematics

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 23 23 18
PDF Downloads 4 4 3