Arithmetic space

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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. I.V. Dolgachev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098