Having given the matter a little bit of thought, here are some things you might want to consider when you undertake your research.
1. The problem is almost certainly due to the viewing device, which will only be displaying the sRGB colour space anyway, not being calibrated. Delta-E and gamma setting will both also have an effect.
2. The monitor on your computer where you prepared the image probably only displays sRGB. That will definitely be the case if you are using a laptop screen, which I do not recommend. Even the best I have seen fall short of what I consider acceptable. If you are processing on a laptop, hook it up to a decent external monitor, preferably on which can display Adobe RGB and has a matte screen, and calibrate that.
3. The colour space setting on your camera only applies to JPEGs.
4. When you shoot Raw, colour space is not a concern until you render the image out of Lightroom. Contrary to popular belief, Lightroom does not use the ProPhoto colour space, but in effect has its own which is similar. I have not investigated, but I would imagine much the same applies to other Raw converters.
5. The colour gamut which digital cameras are capable of recording approximates Adobe RGB. Some colours will fall within its boundaries, others might be just outside. The main benefit of using ProPhoto is that there will be no clipping of any out of gamut colours. How much clipping actually occurs will depend on the medium to which you output. Leaving the image set to ProPhoto avoids the need to have multiple versions.
6. One of the appendices in Martin Evening's excellent Lightroom reference book gives a good explanation of how colour is handled in Lightroom and is a good place to start reading.
Seeing CMYK described as a colour space and rendered on a RGB display certainly made me smile.