The positive-definiteness axiom used for just about all the definitions of inner-product spaces that I've seen goes like this:
$$\langle \mathbf{x},\mathbf{x}\rangle \ge 0 \text{ with equality only for } \mathbf{x}=\mathbf{0}$$
For example here. I'm having trouble rewriting that statement using logical connectives. Is this it?
$$\langle \mathbf{x},\mathbf{x}\rangle = 0 \Leftrightarrow \mathbf{x}=\mathbf{0}$$ and
$$\langle \mathbf{x},\mathbf{x}\rangle > 0 \Leftrightarrow \mathbf{x}\neq\mathbf{0}$$
I feel that my second statement can be scrapped, since it is basically the contrapositive of the first. (I was looking for a "minimal axiom" in the sense of assuming the least.)
In general, I feel that "property $P$ holds only for $x$" means "$x$ iff $P$". Am I right?
This seems to depend on exactly what you take to be in your logical language. Also, you're going to need some quantifiers if you want to say that this holds in general.
Let $I_{x y}$ denote the inner product operation $\langle x, y \rangle$. Then we have that $$\forall x (I_{x x} \ge 0 \wedge (I_{xx} = 0 \Leftrightarrow x=0))$$