If you are truly only interested in the triangle inequality, then there is the Hamming weight
$w(x_1,x_2,\ldots,x_n)=m$ where $m$ is simply the number of non-zero components. This gives you a metric. Mind you, the space $F^n$ is finite, so any metric on it is going to give you the discrete topology. Adding any kind of norm-like requirements (on top of the triangle inequality) is problematic for several reason, as others have pointed out.

The Hamming weight obviously depends on the choice of basis, which may restrict its usefulness (depending on what you wanted to do with this 'norm').

I'm sad to say there isn't an awful lot of analysis happening in this space.

Real norms on vector spaces over finite fields
I've noticed some pain killers working great for me, while others have no effect.

Works for me


Aspirin
APC &dagger;
Naproxen


Doesn't work for me


Paracetamol
Diclofenac
Tramadol


I doubt there is much of a placebo effect at work, since most of these either did or did not work when I first took them, without having expectations either way.

Whenever I have a head ache, I take an APC. I suspect it's actually the aspirin in there that does the job, since when I take just paracetamol, it doesn't do squat. As a kid I got children's aspirin, which worked.

I once had a severe back ache. I was prescribed diclofenac (a heavier variant than the over the counter one), which didn't work. I was then prescribed tramadol &mdash; same results. I then tried naproxen, which worked rightaway.

Why do some pain killers work while others don't?

Is there an underlying mechanism, that explains why some of these work while others don't? Does that predict if pain killers that I haven't used yet will work?

Please note that I'm not looking for medical advice on which pain killers to take; I'm just curious about how my body interacts with the various ones.



&dagger;: the one consisting of aspirin, paracetamol, and caffeine, not the one containing phenaticin. Think Excedrin.
