Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

**Elzbieta Kraszewska**- Warsaw University, Bialystok
**Jan Popiolek**- Warsaw University, Bialystok

- In [14] the series of real numbers were investigated. The introduction to Banach and Hilbert spaces ([10], [11],[12]), enables us to arrive at the concept of series in Hilbert space. We start with the notions: partial sums of series, sum and $n$-th sum of series, convergent series (summable series), absolutely convergent series. We prove some basic theorems: the necessary condition for a series to converge, Weierstrass' test, d'Alembert's test, Cauchy's test.

Contents (PDF format)

- [1]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Jaroslaw Kotowicz.
Convergent sequences and the limit of sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Jaroslaw Kotowicz.
Monotone real sequences. Subsequences.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Jan Popiolek.
Some properties of functions modul and signum.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Jan Popiolek.
Real normed space.
*Journal of Formalized Mathematics*, 2, 1990. - [10]
Jan Popiolek.
Introduction to Banach and Hilbert spaces --- part I.
*Journal of Formalized Mathematics*, 3, 1991. - [11]
Jan Popiolek.
Introduction to Banach and Hilbert spaces --- part II.
*Journal of Formalized Mathematics*, 3, 1991. - [12]
Jan Popiolek.
Introduction to Banach and Hilbert spaces --- part III.
*Journal of Formalized Mathematics*, 3, 1991. - [13]
Konrad Raczkowski and Andrzej Nedzusiak.
Real exponents and logarithms.
*Journal of Formalized Mathematics*, 2, 1990. - [14]
Konrad Raczkowski and Andrzej Nedzusiak.
Series.
*Journal of Formalized Mathematics*, 3, 1991. - [15]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [16]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989.

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