Volume 13, 2001

University of Bialystok

Copyright (c) 2001 Association of Mizar Users

**Ewa Gradzka**- University of Bialystok

- In this paper we define the algebra of formal power series and the algebra of polynomials over an arbitrary field and prove some properties of these structures. We also formulate and prove theorems showing some general properties of sequences. These preliminaries will be used for defining and considering linear functionals on the algebra of polynomials.

This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

- Preliminaries
- The Algebra of Formal Power Series
- The Algebra of Polynomials

- [1]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Jaroslaw Kotowicz.
Functions and finite sequences of real numbers.
*Journal of Formalized Mathematics*, 5, 1993. - [7]
Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski.
Abelian groups, fields and vector spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Robert Milewski.
Fundamental theorem of algebra.
*Journal of Formalized Mathematics*, 12, 2000. - [9]
Robert Milewski.
The ring of polynomials.
*Journal of Formalized Mathematics*, 12, 2000. - [10]
Michal Muzalewski.
Construction of rings and left-, right-, and bi-modules over a ring.
*Journal of Formalized Mathematics*, 2, 1990. - [11]
Michal Muzalewski and Wojciech Skaba.
From loops to abelian multiplicative groups with zero.
*Journal of Formalized Mathematics*, 2, 1990. - [12]
Michal Muzalewski and Leslaw W. Szczerba.
Construction of finite sequence over ring and left-, right-, and bi-modules over a ring.
*Journal of Formalized Mathematics*, 2, 1990. - [13]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Jan Popiolek.
Real normed space.
*Journal of Formalized Mathematics*, 2, 1990. - [15]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [16]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [17]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [18]
Wojciech A. Trybulec.
Groups.
*Journal of Formalized Mathematics*, 2, 1990. - [19]
Wojciech A. Trybulec.
Pigeon hole principle.
*Journal of Formalized Mathematics*, 2, 1990. - [20]
Wojciech A. Trybulec.
Subspaces and cosets of subspaces in vector space.
*Journal of Formalized Mathematics*, 2, 1990. - [21]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [22]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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