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Arithmetic space

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number space, coordinate space, real $ n $- space

A Cartesian power $ \mathbf R ^ {n} $ of the set of real numbers $ \mathbf R $ having the structure of a linear topological space. The addition operation is here defined by the formula:

$$ ( x _ {1} \dots x _ {n} ) + ( x _ {1} ^ \prime \dots x _ {n} ^ \prime ) = ( x _ {1} + x _ {1} ^ \prime \dots x _ {n} + x _ {n} ^ \prime ); $$

while multiplication by a number $ \lambda \in \mathbf R $ is defined by the formula

$$ \lambda ( x _ {1} \dots x _ {n} ) = \ ( \lambda x _ {1} \dots \lambda x _ {n} ). $$

The topology in $ \mathbf R ^ {n} $ is the topology of the direct product of $ n $ copies of $ \mathbf R $; its base is formed by open $ n $- dimensional parallelepipeda:

$$ I = \{ {( x _ {1} \dots x _ {n} ) \in \mathbf R ^ {n} } : { a _ {i} < x _ {i} < b _ {i} , i = 1 \dots n } \} , $$

where the numbers $ a _ {1} \dots a _ {n} $ and $ b _ {1} \dots b _ {n} $ are given.

The real $ n $- space $ \mathbf R ^ {n} $ is also a normed space with respect to the norm

$$ \| x \| = \sqrt {x _ {1} ^ {2} + \dots +x _ {n} ^ {2} } , $$

where $ x = ( x _ {1} \dots x _ {n} ) \in \mathbf R ^ {n} $, and is a Euclidean space with respect to the scalar product

$$ \langle x, y \rangle = \sum _ {i=1 } ^ { n } x _ {i} y _ {i} , $$

where $ x = ( x _ {1} \dots x _ {n} ) , y = ( y _ {1} \dots y _ {n} ) \in \mathbf R ^ {n} $.

How to Cite This Entry:
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