The problem: Given a vector space $V$ with scalars in $\mathbb{R}$, and a positive-definite bilinear form $A$, show $N(v)=(A(v,v))^\frac12$ defines a norm on $V$.

I have easily handled everything but the inequality requirement:

$N(u + v) \leq N(u) + N(v)$

From searching around I have found the hint to use the Cauchy-Schwarz inequality:

$(\sum u\_i v\_i)^2 \leq (\sum u\_i^2)(\sum v\_i^2)$

But I still don't see it.

If I take $A$ to be a dot product, an example of a function that satisfies the definition, I can finally do it:

$N(u + v)^2 = \sum (u\_i + v\_i)^2 = \sum u\_i^2 + 2u\_i v\_i + v\_i^2$

$(N(u) + N(v))^2 = N(u)^2 + 2 N(u) N(v) + N(v)^2 = \sum u\_i^2 + 2 \sqrt{\sum u\_i^2 \sum v\_i^2} + \sum v\_i^2$

Subtracting and dividing out the common parts and squaring it fits right into Cauchy-Schwarz.

However I don't know how to do this if I can't assume $A(u,v) = u \cdot v$