Given a vector $x \in \mathbb{R}^n$ and a finite collection of vectors $ G \subset\mathbb{R}^n $.
are the two conditions sufficient and necessary for each other?
A: $\forall g \in G,\langle x, g \rangle >0 $
and B: $\forall S \subset G, \sum_{i\in S} \langle i, x \rangle > 0$
It's rather easy to prove $A \to B$, since $S \subset G, i \in G$ any finite sum of positive is still positive.
And I'm looking for help to prove $B \to A$.
actually, the proof of $B \to A$ is also easy.
Since $\forall S \subset G$, we have a positive result.
We can construct a set $\{\{g\}|\forall g \in G\}$, which would then prove $A$