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 $2^{2^k}$.
It's easy to get the recurrence $S(u^2) = (1+u) S(u) + \Theta(u)$. (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 (and others) gave in [1] the trivial bound $S(u) = O(u \log \log u)$, and later in [2] he found a clever way to combine the data structure with another one in order to reach space complexity $O(u)$, while maintaining 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. (ZBL1187.68679) leaves it as an exercise.
I tried with some friends to [dis]prove the $O(u)$ bound without luck.
van Emde Boas, P.; Kaas, R.; Zijlstra, E., Design and implementation of an efficient priority queue, Math. Syst. Theory 10(1976), 99–127 (1977). ZBL0363.60104.
van Emde Boas, P., Preserving order in a forest in less than logarithmic time and linear space, Inf. Process. Lett. 6, 80–82 (1977). ZBL0364.68053.