Arithmetic space
From Encyclopedia of Mathematics
number space, coordinate space, real 
-space
A Cartesian power 
of the set of real numbers 
having the structure of a linear topological space. The addition operation is here defined by the formula:
 |
while multiplication by a number 
is defined by the formula
 |
The topology in 
is the topology of the direct product of 
copies of 
; its base is formed by open 
-dimensional parallelepipeda:
 |
where the numbers 
and 
are given.
The real 
-space 
is also a normed space with respect to the norm
 |
where 
, and is a Euclidean space with respect to the scalar product
 |
where 
.
How to Cite This Entry:
Arithmetic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_space&oldid=45223
Arithmetic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_space&oldid=45223
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article