We introduce the Einstein notation, often used by physicist. It is given a vector space, for example and a real metric . Remark that a metric can be assigned once a real matrix is given. In physics, it is written:
equivalent writings are:
then:
We say that repeated indeces are “saturated”. Notice that for being saturated, the repeated indeces cannot be both up or down. This notation is particolary usefull for writing tensors. Let be a vector space over a field with an inner product non degenerate. Recall that a tensor is a multilinear application:
In index notation, is written as with each taking values in . By saturating with vectors, we obtain a scalar in . Furthermore, moving upside down an index, i.e. multiplying for the metric , has now the obvious meaning of substituting a term in the tensor product with the dual, since a non degenerate inner product makes the vector space isomorphic to its dual.