Let {Xnk} be an array of rowwise conditionally independent
random elements in a separable Banach space of type p, 1≤p≤2.
Complete convergence of n−1r∑k=1nXnk to 0, 0<r<p≤2 is obtained
by using various conditions on the moments and conditional means. A
Chung type strong law of large numbers is also obtained under suitable
moment conditions on the conditional means.