We call S(u) the space complexity of the vEB tree holding elements in the range 0 to u-1, and suppose without loss of generality that u is of the form 22k.
It's easy to get the recurrence S(u2) = (1+u) S(u) + Θ(u). (In Wikipedia's article the last term is O(1), but it's wrong because we must count the space for the array.)
Van Emde Boas gave in  the trivial bound S(u) = O(u log log u), and later in  he found a clever way to combine the data structure with other one so that to reach space complexity O(u), while mantaining the O(log log u) time bounds.
But modern references present the original data structure and state without proof that the space complexity is O(u). For instance, the very recent 3rd edition of "Introduction to algorithms" by Cormen et al. leaves it as an exercise.
I tried with some friends to [dis]prove the O(u) bound without luck.