The entrance to my bedroom is through a space about 40"w x 50"l. There are closet doors on either side, the bedroom at one end and the door at the other. There's a full length mirror on the back of the door. The only light in this little space is whatever spills over from the room and it's just terrible. I can't really see anything in the mirror; it's all dim and backlit. Time to install a light.

My first thought is a ceiling light of some sort, either a can or a ceiling fixture something like the one in the link, which would echo a ceiling fixture in the bedroom proper.

Another possibility would be to hang something on the wall over the door, so that it shines down over the mirror when the door is closed.

I can't hang lights on the sides of the door, there's just no room.

So my questions are whether I'd be better off with the ceiling light or the light over the door (in which case I guess I'm going for something like you'd put over a mirror in a bathroom?) and if I'm going for a ceiling fixture, what kind and how far back, so that someone looking in the mirror is properly lit?

Here is a link that might be useful: Ceiling light