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:
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while multiplication by a number is defined by the formula
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The topology in is the topology of the direct product of
copies of
; its base is formed by open
-dimensional parallelepipeda:
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where the numbers and
are given.
The real -space
is also a normed space with respect to the norm
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where , and is a Euclidean space with respect to the scalar product
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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