Results 1  10
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34
Metric spaces
 Formalized Mathematics
, 1990
"... Summary. Sequences in metric spaces are defined. The article contains definitions of bounded, convergent, Cauchy sequences. The subsequences are introduced too. Some theorems concerning sequences are proved. ..."
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Cited by 53 (3 self)
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Summary. Sequences in metric spaces are defined. The article contains definitions of bounded, convergent, Cauchy sequences. The subsequences are introduced too. Some theorems concerning sequences are proved.
Banach space of bounded linear operators
 FORMALIZED MATHEMATICS
, 2003
"... On this article, the basic properties of linear spaces which are defined by the set of all linear operators from one linear space to another are described. Especially, the Banach space is introduced. This is defined by the set of all bounded linear operators. ..."
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Cited by 21 (19 self)
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On this article, the basic properties of linear spaces which are defined by the set of all linear operators from one linear space to another are described. Especially, the Banach space is introduced. This is defined by the set of all bounded linear operators.
Introduction to Banach and Hilbert spaces — part I
 Journal of Formalized Mathematics
, 1991
"... Summary. Basing on the notion of real linear space (see [11]) we introduce real unitary ..."
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Cited by 20 (0 self)
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Summary. Basing on the notion of real linear space (see [11]) we introduce real unitary
Little Bezout theorem (factor theorem)
 FORMALIZED MATHEMATICS
, 2004
"... We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let r belong to a commutative ring L and p(x) be a polynomial over L. Then x − r divides p(x) iff p(r) = 0. We also prove some consequences of this theorem like that any non zero po ..."
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Cited by 13 (3 self)
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We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let r belong to a commutative ring L and p(x) be a polynomial over L. Then x − r divides p(x) iff p(r) = 0. We also prove some consequences of this theorem like that any non zero polynomial of degree n over an algebraically closed integral domain has n (non necessarily distinct) roots.
On ordering of bags
 Journal of Formalized Mathematics
"... Summary. We present a Mizar formalization of chapter 4.4 of [8] devoted to special orderings in additive monoids to be used for ordering terms in multivariate polynomials. We have extended the treatment to the case of infinite number of variables. It turns out that in such case admissible orderings ..."
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Cited by 6 (1 self)
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Summary. We present a Mizar formalization of chapter 4.4 of [8] devoted to special orderings in additive monoids to be used for ordering terms in multivariate polynomials. We have extended the treatment to the case of infinite number of variables. It turns out that in such case admissible orderings are not necessarily well orderings. MML Identifier:BAGORDER. WWW:http://mizar.org/JFM/Vol14/bagorder.html
Firstcountable, sequential, and Frechet spaces
 Journal of Formalized Mathematics
, 1998
"... Summary. This article contains a definition of three classes of topological spaces: firstcountable, Frechet, and sequential. Next there are some facts about them, that every firstcountable space is Frechet and every Frechet space is sequential. Next section contains a formalized construction of to ..."
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Summary. This article contains a definition of three classes of topological spaces: firstcountable, Frechet, and sequential. Next there are some facts about them, that every firstcountable space is Frechet and every Frechet space is sequential. Next section contains a formalized construction of topological space which is Frechet but not firstcountable. This article is based on [10, pp. 73–81].
Primitive roots of unity and cyclotomic polynomials
 Journal of Formalized Mathematics
"... Summary. We present a formalization of roots of unity, define cyclotomic polynomials and demonstrate the relationship between cyclotomic polynomials and unital polynomials. ..."
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Summary. We present a formalization of roots of unity, define cyclotomic polynomials and demonstrate the relationship between cyclotomic polynomials and unital polynomials.
The sequential closure operator in sequential and Frechet spaces
 Journal of Formalized Mathematics
, 1999
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