Arithmetic space

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 parallelepiped:

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

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} $.