sniffnoy | Parallelogramity

So. Got the parallelogram law problem. Basically, a big part of the reason I wasn't getting additivity before was that I was convinced it was much harder than it really was, that you couldn't just use parallelogram law and you needed to somehow do it as two inequalities using triangle inequality. Somehow I recalled Tom saying he did it by triangle inequality; probably I just misremembered. Once Lucas told me he did it using just parallelogram law, I got it within a few tries. (Well, also after he pointed out another way you can use parallelogram law (isolate ||v+w||²), after which I got it immediately (though actually not the way he did it, a more direct way (he went through <v_1+v_2,w>=<2v_1,w>+<v_2-v_1,w>))). So it's actually not such a hard problem after all... I actually came very close a few days ago, but I tried to use triangle inequality instead of parallelogram law and didn't see it!

Hm, so since you actually don't use triangle inequality at all, that means that if ||•|| is positive-definite, scalars come out in absolute value, and satisfies the parallelogram law, then it's automatically a norm generated by an inner product and satisfies triangle inequality as well! I doubt that that's actually much use in anything, but it's neat to note. This is wrong; see below.

-Sniffnoy