The corollary below is from Hoffman and Kunze's book, Linear Algebra.
Corollary. If $W$ is a $k$-dimensional subspace of an $n$-dimensional vector space $V$, then $W$ is the intersection of $(n-k)$ hyperspaces in $V$.
In the proof, they find $n-k$ linear functionals $f_{i}$ such that $W=\cap_{i=1}^{n-k}\ker (f_{i})$.
I want to know if the following is true: For all proper subspace $W$ of a infinite-dimensional vector space $V$ there are a set of linear functionals $\{f_{i}|i\in I\}$, where $I$ is a set of index such that $W=\cap_{i\in I}\ker (f_{i})$.
Thanks for your kindly help.