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
Arithmetic space. I.V. Dolgachev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Arithmetic_space&oldid=13139