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 R n. We also give the standard basis of such spaces. For the set 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. We use the following convention: i, j, n are elements of N, z, B 0 are sets, and f , x 0 are real-valued finite sequences. Next we state several propositions: (1) For all functions f , g holds dom(f · g) = dom g ∩ g −1 (dom f). (2) For every binary relation R and for every set Y such that rng R ⊆ Y holds R −1 (Y) = dom R.