This paper presents a novel training-based adaptive linear receiver to suppress the multiple access interference, which has been one of the central issues in DS/CDMA wireless communication systems. The proposed receiver is derived by extending the Krylov-proportionate normalized least-mean-square (KPNLMS) algorithm to complex-valued signals. The key idea of KPNLMS is (a) sparse representation of the minimum mean square error (MMSE) receiver based on a certain Krylov subspace, (b) use of the 'sparsity' for fast convergence, and (c) simplification to keep linear complexity per iteration. To clarify the convergence properties of the proposed receiver, its derivation from the variable-metric version of the adaptive projected subgradient method (V-APSM) is presented. The simulation results demonstrate the efficacy of the proposed receiver.