1. A computer program product deployed in a storage controller for implementing a space efficient volume in a storage system for a host system transmitting writes to the space efficient volume, wherein the computer program product comprises a computer readable storage medium having program instructions embodied therewith, the program instructions executable by a processor to cause operations in the storage controller, the operations comprising: maintaining the space efficient volume in the storage system, wherein physical storage space is allocated to the space efficient volume when the host system submits a write request, wherein an amount of space needed by the write request is allocated to the space efficient volume in response to receiving the write request, wherein the storage controller and the host system comprise separate computing systems; receiving, from the host system, a point-in-time copy establish request to create a point-in-time copy relationship to copy source data to target data, wherein the point-in-time copy establish request includes a space pre-allocation 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.