A simple example of a multilinear function is the inner (a.k.a dot) product <a, b>: it takes a vector (b), and a dual vector (a^T), and returns a number. In ten

A simple example of a multilinear function is the inner (a.k.a dot) product <a, b>: it takes a vector (b), and a dual vector (a^T), and returns a number. In tensor notation it's typically written δ_ij. It's multilinear because it's linear in each of its arguments separately: <ca, b> = c<a,b> and <a, cb> = c<a,b>. Another simple but less obvious example is a rotation (orthogonal) matrix. It takes a vector as an input, and returns a vector. But a vector itself can be thought of as a linear function that takes a dual vector and returns a number (via the inner product, above!). So, applying the rotation matrix to a vector is a sort of "currying" on the multilinear map, while the matrix alone can be considered a function that takes a vector and a dual vector, and returns a number. In functional notation, you can consider your rotation matrix to be a function (V x V*) -> K, which can in turn be considered a function V -> (V* -> K), where V* is the dual space of V.