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We study composition operators on the Schwartz space of rapidly decreasing functions. We prove that such a composition operator is never a compact operator and we obtain necessary or sufficient conditions for the range of the composition operator to be closed. These conditions are expressed in terms of multipliers for the Schwartz class and the closed range property of the corresponding operator considered in the space of smooth functions. Dedicated to our friend Pepe Bonet on the occasion of his 60th birthdaydoi:10.4171/rmi/989 fatcat:4wrbvhh2dneepflymje6wepmoi