Not trying to be stubborn here - I just don't understand. So we're talking about the case when tensors don’t need to be multilinear functions on a product of v

Not trying to be stubborn here - I just don't understand. So we're talking about the case when tensors don’t need to be multilinear functions on a product of vector spaces. Each element of V* is (trivially) a tensor and a linear function on V. Each element of V is (trivially) a tensor and a linear function on V*. However, not all linear functions on V* are in V. So V** is bigger than V. No problem so far. But all elements of V** are functions on (trivial) products of vector spaces, and by definition all functions in V** are linear. So how have we misunderstood each other here?