Suppose that is a -space for some infinite field and . Now, we may assume WLOG that for any . It follows that there exists with and that there is some . So, consider the set (vectors of the form for ). By assumption we have that is infinite. That said, note that (since otherwise ) and for , for if and were both in then their difference and so . It follows then that can contain at most elements of which is ridiculous since is infinite and .