I wrote an article for the Photographic Society of America about using the ProPhoto colour space in Adobe Photoshop and Lightroom to preserve all the colour that your camera can capture in an image. These are supplemental graphs to show the relationship between the various common colour spaces. As you can see in the graph below ProPhoto is much larger than AdobeRGB and sRGB. If you convert the RAW images from a camera to AdobeRGB before printing you have lost some of the colours that a good quality printer can print. This will limit the colour gamut that you achieve in a print.
Comparing ProPhoto AdobeRGB and sRGB colour spaces.
Comparing ProPhoto, AdobeRGB and Epson 3880 Premium Lustre paper. You print some colours on an Epson 3880 using Premium Lustre paper that are available in the AdobeRGB colour space.