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The question is: Suppose $v_1,...,v_n$ is a linearly independent list in V. Show that there exists $w \in V$ such that $⟨w,v_j⟩ \gt 0$ for all $j \in {1,...,m}$.

From what I understand, solving the question requires the use of the Riesz Representation Theorem. What linear functional would I use and how would I use it?

Thanks.

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    So is $V$ a Hilbert space? – 2017-01-25

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Let $(V, \left< \cdot, \cdot \right>)$ be an inner product space over $\mathbb{R}$ or $\mathbb{C}$ and consider $W := \operatorname{ span }(v_1, \dots, v_n)$. Define a linear functional $\varphi \colon W \rightarrow \mathbb{F}$ by declaring that $\varphi(v_i) = 1$ for all $1 \leq i \leq n$. Since $(W, \left< \cdot, \cdot \right>)$ is finite dimensional, the Riesz representation theorem gives us a vector $w \in W \subseteq V$ such that $\varphi(u) = \left< u, w \right>$ for all $u \in W$. Then

$$ 1 = \varphi(v_i) = \left< v_i, w \right> = \left< w, v_i \right> > 0 $$

for all $1 \leq i \leq n$.