For any natural number n, the set R consists of all n-tuples of real numbers (R). It is called the "n-dimensional real space" … See more

Any function f(x1, x2, ..., xn) of n real variables can be considered as a function on R (that is, with R as its domain). The use of the real n-space, instead of several variables considered separately, can simplify not… See more

The coordinate space R forms an n-dimensional vector space over the field of real numbers with the addition of the structure of linearity, and is often still denoted R . The operations on R as a vector space are typically de… See more

The fact that real numbers, unlike many other fields, constitute an ordered field yields an orientation structure on R . Any full-rank linear map of R to itself either preserves or reverses orientation of the space depending … See more

The topological structure of R (called standard topology, Euclidean topology, or usual topology) can be obtained not only from Cartesian product. It is also identical to the natural topology induced by Euclidean metric disc… See more

Cases of 0 ≤ n ≤ 1 do not offer anything new: R is the real line, whereas R (the space containing the empty column vector) is a singleton, understood as a zero vector space. However, it is useful to include these as … See more