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