A process for producing methanol from a synthesis gas containing hydrogen, carbon oxides and optionally inert components, comprising the following steps: (a) providing a synthesis gas make-up gas stream containing hydrogen, carbon oxides and optionally inert components, which contains a hydrogen deficit with respect to the stoichiometry specified for the methanol synthesis, (b) combining the synthesis gas make-up gas stream deficient in hydrogen with a trimming gas stream containing hydrogen to obtain a synthesis gas make-up gas stream enriched in hydrogen, (c) combining the synthesis gas make-up gas stream enriched in hydrogen with the synthesis gas cycle gas stream obtained in step (g) to obtain a synthesis gas feed gas stream, (d) introducing the synthesis gas feed gas stream into at least one methanol synthesis reactor and at least in part catalytically converting the carbon oxides contained in the synthesis gas feed gas stream to methanol by using hydrogen under methanol synthesis conditions, (e) discharging a product gas stream containing methanol vapor and non-converted synthesis gas constituents from the at least one methanol synthesis reactor, (f) at least partly separating the methanol from the product gas stream by cooling and condensation, (g) fractionating the product gas stream after separation of the methanol into a synthesis gas cycle gas stream and a purge gas stream loaded with methanol residues, recirculating the synthesis gas cycle gas stream to step (c), (h) introducing the purge gas stream loaded with methanol residues into a washing device, contacting the purge gas stream loaded with methanol residues in the washing device with a washing agent, preferably water, discharging a purge gas stream depleted of methanol and a washing agent loaded with methanol from the washing device, (i) introducing the purge gas stream depleted of methanol into a hydrogen separation device which comprises a membrane separation stage and a pressure swing adsorption stage, wherein at least one gas stream enriched in hydrogen and at least one gas stream depleted of hydrogen is obtained, (j) recirculating at least one gas stream enriched in hydrogen as trimming gas stream containing hydrogen to step (b). (a) providing a synthesis gas make-up gas stream containing hydrogen, carbon oxides and optionally inert components, which contains a hydrogen deficit with respect to the stoichiometry specified for the methanol synthesis, (b) combining the synthesis gas make-up gas stream deficient in hydrogen with a trimming gas stream containing hydrogen to obtain a synthesis gas make-up gas stream enriched in hydrogen, (c) combining the synthesis gas make-up gas stream enriched in hydrogen with the synthesis gas cycle gas stream obtained in step (g) to obtain a synthesis gas feed gas stream, (d) introducing the synthesis gas feed gas stream into at least one methanol synthesis reactor and at least in part catalytically converting the carbon oxides contained in the synthesis gas feed gas stream to methanol by using hydrogen under methanol synthesis conditions, (e) discharging a product gas stream containing methanol vapor and non-converted synthesis gas constituents from the at least one methanol synthesis reactor, (f) at least partly separating the methanol from the product gas stream by cooling and condensation, (g) fractionating the product gas stream after separation of the methanol into a synthesis gas cycle gas stream and a purge gas stream loaded with methanol residues, recirculating the synthesis gas cycle gas stream to step (c), (h) introducing the purge gas stream loaded with methanol residues into a washing device, contacting the purge gas stream loaded with methanol residues in the washing device with a washing agent, preferably water, discharging a purge gas stream depleted of methanol and a washing agent loaded with methanol from the washing device, (i) introducing the purge gas stream depleted of methanol into a hydrogen separation device which comprises a membrane separation stage and a pressure swing adsorption stage, wherein at least one gas stream enriched in hydrogen and at least one gas stream depleted of hydrogen is obtained, (j) recirculating at least one gas stream enriched in hydrogen as trimming gas stream containing hydrogen to step (b).