# The Euclidean Space

FORMALIZED MATHEMATICS

Vol.2, No. 4, September–October1991

Universit´e Catholique de Louvain

The Euclidean Space

Agata Darmochwa l

Warsaw University

Bia l ystok

Summary. The general deﬁnitionof Euclidean Space.

MML Identiﬁer:EUCLID .

The papers [14],[6],[9],[8],[12],[1],[5],[10],[3],[13],[4],[15],[16],[7],[11],and [2]provide the notation and terminology for this paper. In the sequel k , n denote natural numbers and r denotes a real number. Let us consider n . The functor R n yields a non-empty set of ﬁnitesequences of

and is deﬁnedas follows:

(Def.1)R n =n .

In the sequel x will denote a ﬁnitesequence of elements of . The function ||from into is deﬁnedas follows:

(Def.2)for every r holds ||(r ) =|r |.

Let us consider x . The functor |x |yields a ﬁnitesequence of elements of

and is deﬁnedas follows:

(Def.3)|x |=||·x .

Let us consider n . T
he functor 0, ... , 0 n yields a ﬁnitesequence of elements of and is deﬁnedby:

Let us consider n . The functor 0, ... , 0 n yields a ﬁnitesequence of elements of and is deﬁnedby:

(Def.4) 0, ... , 0

n =n −→0qua a real number . Let us consider n . Then 0, ... , 0

n is an element of R n .

In the sequel x , x 1, x 2, y denote elements of R n . One can prove the following proposition

(1)x is an element of n . 601c 1991Fondation Philippe le Hodey

ISSN 0777–4028

Word文档免费下载：

（共5页）

## TOP相关主题

A Covariance Approximation on Euclidean Space for Visual Tracking

A Covariance Approximation on Euclidean Space for Visual Tracking Stefan Kluckner, Thomas Mauthner, and Horst Bischof Institute for Computer Graphics and ...

ON SPACELIKE AUSTERE SUBMANIFOLDS IN PSEUDO-EUCLIDEAN SPACE

ON SPACELIKE AUSTERE SUBMANIFOLDS IN PSEUDO-EUCLIDEAN SPACE_专业资料。 ACTA ATHEMATIMCA SCIENTIA Vb1 e..SrB 31 I hi oentsnt,wegneaieBrn ...

Gauge Model With Extended Field Transformations in Euclidean Space

Let F be a framework in the Euclidean d-space E

Key Words: In nitesimal exibility, exible polyhedra, octahedra MSC 1994: 53A17 1 Introduction Let F be a framework in the Euclidean d-space E with ...

Minimal Surfaces in Euclidean Space Lecture Notes, UAB, Spring

Minimal Surfaces in Euclidean Space Lecture Notes, UAB, Spring Minimal Surfaces in Euclidean Space Lecture Notes, UAB, SpringMinimal Surfaces in Euclidean ...

Hausdorff operators on Euclidean spaces

App1.Math.J.Chinese Univ 2013,28(4):548—564 Hausdorf operators on Euclidean spaces CHEN Jie—cheng FAN Da—shan WANG Si—lei。 Dedicated to ...

Lines in n-dimensional Euclidean spaces

Lines in n-dimensional Euclidean spaces Summary. In this paper, we define the line of n-dimensional Euclidian space and we introduce basic properties of...

(L^p, L^q)-Boundedness of Hausdorff Operators with Power Weight on Euclidean Spaces

(L^p, L^q)-Boundedness of Hausdorff Operators with Power Weight on Euclidean Spaces_电子/电路_工程科技_专业资料。Analysis in Theory and Applications ...

An Addressing and Routing Scheme Based on Modified Euclidean Space for Hexagonal Networks

An Addressing and Routing Scheme Based on Modified Euclidean Space for Hexagonal Networks_电子/电路_工程科技_专业资料。NETWORK TECHNOLOGY AND APPLICATlON An ...

Minimal Surfaces in the Four-Dimensional Euclidean Space

Minimal Surfaces in the Four-Dimensional Euclidean Space We prove that the Gauss curvature and the curvature of the normal connection of any minimal ...