In this paper, we prove a theorem on convergence in mean of order p for double arrays of pairwise independent random vectors in Hilbert Space, where 1 ≤ p < 2. The main result extends Theorem 2.1 of Bao et al. [2] and Theorem 2.1 of Thanh [8]. The proof is based on the von Bahr–Essen inequality for pairwise independent random vectors taking values in Hilbert spaces.