I’ve tried to give (see on the post) a different description of an equivalence relation that I find intuitive and I think gives the space V as we want it, but it may not be fully correct.
I haven’t read too closely, but it looks like the equivalence relation that you’re talking about in the post sets elements that are scalar multiples of each other in equivalence. This isn’t the point of my equivalence; the stuff I wrote is all in terms of vectors, not directions. My other top-level comment discusses this.
I’ve tried to give (see on the post) a different description of an equivalence relation that I find intuitive and I think gives the space V as we want it, but it may not be fully correct.
I haven’t read too closely, but it looks like the equivalence relation that you’re talking about in the post sets elements that are scalar multiples of each other in equivalence. This isn’t the point of my equivalence; the stuff I wrote is all in terms of vectors, not directions. My other top-level comment discusses this.