The space hierarchy theorem guarantees that DSPACE(logd n) ⊊ DSPACE(logd + 1 n) for all integers d > 0. If polyL had a complete problem, call it A, it would be an element of DSPACE(logk n) for some integer k > 0. Suppose problem B is an element of DSPACE(logk + 1 n) \ DSPACE(logk n). The assumption that A is complete implies the following O(logk n) space algorithm for B: reduce B to A in logarithmic space, then decide A in O(logk n) space. This implies that B is an element of DSPACE(logk n) and hence violates the space hierarchy theorem.
