Euclidean Space
Euclidean 
-space, sometimes called Cartesian space
or simply 
-space, is the space
of all n-tuples of real
numbers, (
, 
, ..., 
). Such 
-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 
-vectors. 
is the set of real
numbers (i.e., the real line), and 
is called the
Euclidean plane. In Euclidean space, covariant
and contravariant quantities are equivalent
so 
.
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