Show that PTIME and PSPACE is closed under Klenee star
Suppose you are given a language $L$ in one of these complexity classes, and you want to decide $L^*$ in the same complexity class. On input $x_1 \ldots x_n$ (where the $x_i$ are bits), use dynamic programming to find out which substrings $x_i x_{i+1} \ldots x_j$ belong to $L^*$. Since there are only $O(n^2)$ such substrings, this will only cause a polynomial blow-up in the resources used.