A lot of elementary school gyms have carpet in them now and we were thinking of doing the same thing in our basement. It looks like they use a very low pile carpet and no pad. In the office I work at they have basically the same thing and the carpet is glued to the concrete. I was wondering if you could use carpet tape instead to make it easier to change the floor later.
If you're even thinking about gluing carpet down with no pad, consider paying a little bit extra for an industrial grade carpet similar to what they use in large airports and office buildings.
My wife and I did this and it held up for over 15 years under normal usage.
It's even better if it's an indoor/outdoor carpet. Most of these come with a bit of rubber backing on them, so you get a little bit of padding.
One reason I suggest this upgrade is that regular carpet without a pad does not wear as well as carpet with a pad.