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''number space, coordinate space, real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a0133802.png" />-space''
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$#C+1 = 19 : ~/encyclopedia/old_files/data/A013/A.0103380 Arithmetic space,
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A Cartesian power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a0133803.png" /> of the set of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a0133804.png" /> having the structure of a linear topological space. The addition operation is here defined by the formula:
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{{TEX|auto}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a0133805.png" /></td> </tr></table>
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''number space, coordinate space, real  $  n $-
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space''
  
while multiplication by a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a0133806.png" /> is defined by the formula
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A Cartesian power  $  \mathbf R  ^ {n} $
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of the set of real numbers  $  \mathbf R $
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having the structure of a linear topological space. The addition operation is here defined by the formula:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a0133807.png" /></td> </tr></table>
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$$
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( x _ {1} \dots x _ {n} ) + ( x _ {1}  ^  \prime  \dots x _ {n}  ^  \prime  )
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= ( x _ {1} + x _ {1}  ^  \prime  \dots x _ {n} + x _ {n}  ^  \prime  );
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$$
  
The topology in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a0133808.png" /> is the topology of the direct product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a0133809.png" /> copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a01338010.png" />; its base is formed by open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a01338012.png" />-dimensional parallelepipeda:
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while multiplication by a number  $  \lambda \in \mathbf R $
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is defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a01338013.png" /></td> </tr></table>
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$$
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\lambda ( x _ {1} \dots x _ {n} )  = \
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( \lambda x _ {1} \dots \lambda x _ {n} ).
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$$
  
where the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a01338014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a01338015.png" /> are given.
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The topology in  $  \mathbf R  ^ {n} $
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is the topology of the direct product of  $  n $
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copies of  $  \mathbf R $;
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its base is formed by open  $  n $-
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dimensional parallelepipeda:
  
The real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a01338016.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a01338017.png" /> is also a normed space with respect to the norm
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$$
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= \{ {( x _ {1} \dots x _ {n} ) \in \mathbf R  ^ {n} } : {
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a _ {i} < x _ {i} < b _ {i} , i = 1 \dots n } \}
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,
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a01338018.png" /></td> </tr></table>
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where the numbers  $  a _ {1} \dots a _ {n} $
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and  $  b _ {1} \dots b _ {n} $
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are given.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a01338019.png" />, and is a Euclidean space with respect to the scalar product
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The real  $  n $-
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space  $  \mathbf R  ^ {n} $
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is also a normed space with respect to the norm
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a01338020.png" /></td> </tr></table>
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$$
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\| x \|  = \sqrt {x _ {1}  ^ {2} + \dots +x _ {n}  ^ {2} } ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013380/a01338021.png" />.
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where $  x = ( x _ {1} \dots x _ {n} ) \in \mathbf R  ^ {n} $,
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and is a Euclidean space with respect to the scalar product
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$$
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\langle  x, y \rangle  =  \sum _ {i=1 } ^ { n }
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x _ {i} y _ {i} ,
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$$
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 +
where  $  x = ( x _ {1} \dots x _ {n} ) , y = ( y _ {1} \dots y _ {n} ) \in \mathbf R  ^ {n} $.

Revision as of 18:48, 5 April 2020


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=13139
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article