I was reading about hyperplanes online and came across a text that said that every subspace of V is the intersection of hyperplanes. (A hyperplane in V is defined as the kernel of a linear functional.)
I found this fact to be interesting and surprising, so I'm trying to find a detailed proof, however, cannot find one. Please can someone provide me with a clear proof? I'm very new to dual spaces, thanks.
Suppose $U$ is a closed subspace of the Hilbert space $V$. Let $\{e_j : j \in J\}$ be an orthonormal basis for $U^{\perp}$. Then $$U = (U^{\perp})^{\perp} = \{v \in V : (v, e_j) = 0 \text{ for all } j \in J\} = \bigcap_{j \in J}\ker e_j^*,$$ where $e_j^* : V \to \mathbb{C}$ is the linear functional $e_j^*v = (v, e_j)$.
In applications it seems that $V$ is usually separable, so that $J$ is at most countable.