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Let p≥1/α and 1/2<α≤1. Let {X,Xn,  n≥1} be a sequence of independent and identically distributed B-valued random elements and let {ani,  1≤i≤n,  n≥1} be an array of real numbers satisfying ∑i=1naniq=O(n) for some q>p. We give necessary and sufficient conditions for complete moment convergence of the form ∑n=1∞n(p-v)α-2E∑i=1naniXi-εnα+v<∞,  ∀ε>0, where 0<v<p. A strong law of large numbers for weighted sums of independent B-valued random elements is also obtained.