Let $E$ be an infinite-dimensional vector space and $\left(f_{n}\right)_{n \in \mathbb{N}}$ a family of linear forms on $E$. We denote
$$K = \bigcap_{n \in \mathbb{N}} \operatorname{Ker}\left(f_{n}\right)$$
Show that the codimension of $K$ in $E$ is equal to the rank of the family $\left(f_{n}\right)_{n \in \mathbb{N}}$ in the dual space $E^{*}$ (begin with the case where this rank is finite).