I got stuck in a math problem i recently got in functional analysis:
It's some kind of Lemma of Farkas but i can't find it somewhere in the Internet:
Let X be a real locally convex space and $\xi, \xi_{1},...,\xi_{n} \in X^{'}$ , such that if x ∈ X with $\xi_{i}(x) \ge 0$ for all $i = 1, . . . , n$, then $\xi(x) \ge 0$. Show the existence of $a_i \ge 0$ such that $\xi = \sum_{i=1}^{n}a_i \xi_i$.
As a hint i got that i have to use a separation theorem (of Hahn Banach) on $\{\sum_{i=1}^{n} a_i \xi_i | a_i \ge 0 \, \text{for } i = 1,...,n \}$.
I would appreciate any kind of help! Thank you!