Let us define a **one-form** as a linear function that takes a vector and maps it to a real number.

What do we mean by linear? We mean that f(**a**+**b**) = f(**a**) + f(**b**) and f(k**a**) = kf(**a**).

Now let us define a **tensor** as a linear function that takes zero or more vectors and zero or more one-forms and maps them to a real number. In particular let us designate a tensor to be of rank (*m*,*n*) if it takes *m* one-forms and *n* vectors.

A tensor of rank (0,0) is just a constant mapping to a real number. So a real number can be viewed as a tensor of rank (0,0).

By our definition of one-form above, a tensor of rank (0,1) is just a one-form.

Now, a tensor of rank (1,0) is a linear mapping from one-forms to the real numbers. For every such mapping, there is a vector that is equivalent to this mapping. If the vector is **v** then the mapping is the mapping that takes the one-form f to the real number f(**v**). In other words, a tensor of rank (1,0) can be thought of as a constant mapping to a vector.

A tensor of rank (0,2) takes two vectors and maps them linearly to a real number. This is actually equivalent to taking mapping from a vector to a one-form. (Does the tensor have to be symmetric for this to be true?) If you give a tensor of rank (0,2) just one vector you have a tensor that needs one (more) vector to make a real number - in other words, you have a one-form. See currying tensors.

A tensor of rank (2,0) takes two one-forms and maps them linearly to a real number. (Is equivalent to saying that a tensor of rank (2,0) is a constant mapping to two vectors?)

A tensor of rank (1,1) can be viewed as a mapping from a one-form and vector to a real number or as a mapping from a vector to a vector or as a mapping from a one-form to a one-form.

In fact, a tensor or rank (*m*, *n*) can be viewed as:

- a mapping from
*m*one-forms and*n*vectors to a real number - a mapping from
*n*vectors to*m*vectors - a mapping from
*m*one-forms to*n*one-forms