It is well known this lemma.
Lemma: Let $E$ a vector space and $\{f_k\}_{k=1}^n\subseteq E^*$. Write $N=\bigcap_{k=1}^n\ker f_k$ Then, $\dim(E/N)\leq n$.
If $n=1$, then $f:E\to\mathbb{K}$ can be factored through $\pi:E\to E/N$ and a monomorphism $f^*:E/N\to\mathbb{K}$ as $f=f^*\circ\pi$. Then, $\dim(E/N)\leq\dim(\mathbb{K})=1$.
Can we do similar stuff in the general case?