Abstract: We obtain general results of complete convergence for array4 of rowwise independent random elements in a separable Banach space which are stochastically bounded by a positive random variable and discuss some convergence results for the sequence of weighted sums of the form ∑_i=1ⁿ a_mX_m,n ≥ 1.Some results in[5],[6]are improved and extended. Meanwhile, We discuss the equivalence of the complete convergence and the properties Banach Space of type p as well. Some results in[14],[15]are further improved.

Key words: Random element in Banach Space, Complete Convergence, Banach Space of type p, Law of large number