1. A computer program product deployed for implementing a space efficient volume in a storage system, wherein the computer program product comprises a computer readable storage medium having program instructions embodied therewith that when execute perform operations, the operations comprising: receiving a point-in-time copy establish request to create a point-in-time copy relationship to copy source data to target data and a space pre-allocation request provided with the point-in-time copy establish request, indicating a requested amount of space to pre-allocate in the space efficient volume for the target data to which the source data is copied as part of the point-in-time copy relationship for the point-in-time copy establish request; creating the point-in-time copy relationship in response to the point-in-time copy establish request; and pre-allocating the requested amount of space in the space efficient volume for the target data in response to the space pre-allocation request in advance of the point-in-time copy relationship needing the requested amount of space in the target data for the source data to be copied for the point-in-time copy relationship.