Euclidean 
-space,
sometimes called Cartesian space or simply 
-space, is the space of all n-tuples
of real numbers, (
, 
, ..., 
). Such n-tuples are sometimes
called points, although other nomenclature may be used
(see below). The totality of 
-space is commonly denoted 
, although older literature uses the symbol 
(or actually, its non-doublestruck variant 
; O'Neill 1966, p. 3).
is a vector
space and has Lebesgue covering dimension 
. For this reason, elements
of 
are sometimes called n-vectors. 
is the set of real
numbers (i.e., the real line), and 
is called the Euclidean plane.
The Euclidean metric identifies covariant
and contravariant quantities. In an orthonormal
basis, the metric-dual vector basis vectors
satisfy 
;
this equality need not hold in an arbitrary vector basis.
