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We have considered semi-linear stochastic delay systems with Lévy noise. First we study Lévy processes in Hilbert space and present the result for Lévy processes in Hilbert space that the processes can be present as the sum of the series. Then we apply the result to investigate the approximate controllability for stochastic delay systems with Lévy noise. The sufficient conditions for approximate controllability results are established for these stochastic delay systems [ABSTRACT FROM PUBLISHER]