I was working on an inner product proof. From what I've learned, the positive-definite axiom of inner products involves showing that:
Given a vector space $V$ over $F$, for all $v \in V$, $\langle v, v \rangle \in R$ and $\langle v, v\rangle \geq 0$ with $\langle v, v \rangle = 0$ iff $v = 0$. Or alternatively, $\langle v, v \rangle \geq 0$ with equality iff $v = 0$.
While the other axioms are fairly easy to comprehend, I'm a little confused about this one. In particular, what does it expect one to prove?
My thoughts: First, show that $\langle v, v \rangle \in R$ and $\langle v, v \rangle \geq 0$ for all $v \in V$. Then, assume $v = 0$ and show that $\langle v, v \rangle = 0$. Then, assume $\langle v, v \rangle = 0$ and show that $v = 0$.
Is that what it requires? Or, is one supposed to assume that $v = 0$ and show that $\langle v, v \rangle \geq 0$ AND $\langle v, v \rangle = 0$, and then, assume $\langle v, v \rangle \geq 0$ and $\langle v, v \rangle = 0$ and show that $v = 0$?
After seeing some proofs, I'm still unsure. Any explanation is much appreciated.