By definition, the scalars of a vector space are its tensors of rank 0.
In anyvector space,a linear function which sends a vector to a scalar may be calleda covector. Normally, covectors and vectors are different types of things. (Think of the bras and kets ofquantum mechanics.) However, if we are considering only finitely many dimensions,then the space of vectors and the space of covectors have thesame number of dimensions and can therefore be put in a linear one-to-one correspondence with each other.
Such a bijectivecorrespondence is called a metric and is fully specified by a nondegenerate quadratic form, denoted by a dot-product ("nondegenerate" precisely means that the associated correspondenceis bijective).
Once a metric is defined, we are allowed to blur completelythe distinction betweenvectors and covectors as they are now in canonical one-to-onecorrespondence. 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).
