Minimal gap between basement ceiling and main floor floor
If the "gap" you speak of is between the floor/ceiling surfaces, it's probably already minimal, as typical floor framing is about that size.

If it's between the ceiling surface and the bottom of the floor framing, you'll have to find out what is running in the gap, where, before knowing if (or at least where) you can raise the ceiling to be attached to the bottom of the floor framing (or some intermediate framing/resilient channel attached to the floor framing.) Commonly there will be some plumbing and/or ducts in the space, for some parts of the basement.