I am working through the finite element book by Ciarlet and am currently looking at the Deny Lion's Lemma (Theorem 3.1.1 p. 115). 

The Lemma essentially wants to show that $\inf\_{p \in P\_{k}}\Vert v - p\Vert\_{k+1,p,\Omega} \leq C \vert v \vert\_{k+1,p,\Omega}$. 

During the proof, we consider linear functionals $f\_{i}$ on $P\_{k}$ and then make use of the Hahn-Banach Theorem to extend these to functionals on $W^{k+1,p}(\Omega)$. The claim is then made that for each $v \in W^{k+1,p}(\Omega)$ we can select a $q \in P\_{k}$ such that $f\_{i}(v-q) = 0$ for all $i$. 

I am struggling to see how we reason this ability to choose such a polynomial $q$. Would anyone be able to point me in the right direction?