A norm on a vector space comes from an inner product if and only if it satisfies the parallelogram law. Given such a norm, one can reconstruct the inner product via the formula:
$2\langle u,v\rangle = |u + v|^2 - |u|^2 - |v|^2$
(there are minor variations on this)
It's straightforward to prove, using the parallelogram law, that this satisfies:
- $\langle u,u\rangle \ge 0$ for all $u$, and $\langle u,u\rangle = 0$ iff $u = 0$
- $\langle tu,tu\rangle = t^2 \langle u,u\rangle$;
- $\langle u,v\rangle = \langle v,u\rangle$;
- $\langle u,v+w\rangle = 2\langle u/2,v\rangle + 2\langle u/2,w\rangle$;
From 4 with the special case $w=0$ one quickly deduces that $\langle u,v+w\rangle = \langle u,v\rangle + \langle u,w\rangle$.
The usual method of proving $\langle u,tv\rangle = t\langle u,v\rangle$ is to use 4 with induction to prove that $\langle u,nv\rangle = n\langle u,v\rangle$, then deduce $\langle u,tv\rangle = t\langle u,v\rangle$ for $t$ rational, and finally appeal to continuity to extend to the reals.
Is there any way to avoid this last bit? In particular, is there a more geometric view of why $\langle u,tv\rangle = t\langle u,v\rangle$ for all real $t$? Pictures would be great!
If not, is there a different way to express the condition that a norm comes from an inner product that does make all the conditions obviously geometrical?
Clarification added later: My reason for asking this is pedagogical. I teach a course which introduces, in quick succession, metric spaces, normed vector spaces, and inner product spaces. The properties of metrics and norms are very easy to motivate from intuitive properties of distances and lengths. I'd like to do the same for inner products in terms of angles. Thus by "geometric" I mean "geometric intuition" rather than geometry as geometers understand it. Since the inner product is introduced after the norm, I argue that using the cosine law one can define the notion of "angle" between two vectors using any norm. However, unless the norm is "special", that notion of angle doesn't behave how we would expect it to do so. In particular, in order for angles to add properly, one needs the norm to satisfy the parallelogram law. Here "add" means that (modulo a pi or two), the angle from $u$ to $v$ plus the angle from $v$ to $w$ should be the angle from $u$ to $w$. Once one has the parallelogram law then the fact that it comes from an inner product follows via the route above. However, the properties of an inner product are not particularly obvious from thinking about properties of angles. So the easier they are to deduce from the parallelogram law, the easier they are to motivate. I consider the route to $\langle u,\lambda v\rangle = \lambda\langle u,v\rangle$ to be a little long. I was hoping someone could shorten it for me.
Alternative, there may be a different starting point than that angles "add". Perhaps some other property, say similarity of certain triangles, that could be used. However, I'd like a single property that would do the lot. I don't want "add" for some properties and "something else" for others. That's too complicated.