By definition, the scalars of a vector space are its
tensors of rank 0.
In any vector space,
a linear function which sends a vector to a scalar may be called
a covector.
Normally, covectors and vectors are different types of things.
(Think of the bras and kets
of quantum mechanics.)
However, if we are considering only finitely many dimensions,
then the space of vectors and the space of covectors have the
same number of dimensions and can therefore be put in a linear
one-to-one correspondence with each other.
Such a bijective
correspondence is called a metric
and is fully specified by a nondegenerate
quadratic form, denoted by a dot-product
("nondegenerate" precisely means that the associated correspondence
is bijective).
A metric is said to be Euclidean if it is "positive definite", which
is to say that V.V is positive for any nonzero
vector V. Euclidean metrics are nondegenerate but
other metrics exist which are nondegenerate in the above sense
without being "definite" (which is to say that V.V
can be zero even when V is nonzero).
Such metrics are perfectly acceptable.
They include the so-called Lorentzian metric of
four-dimensional spacetime, which is our primary concern here.
Once a metric is defined, we are allowed to blur completely
the distinction between
vectors and covectors as they are now in canonical one-to-one
correspondence. A tensor of rank zero is a scalar.
More generally, a tensor of nonzero rank n
(also called nth-rank tensor, or n-tensor)
is a linear function that maps a vector to a tensor of rank n-1.
Such an object is intrinsically defined,
although it can be specified by
either its covariant or
its contravariant coordinates in a given basis
(cf. 2D example).
