Let $X = C_{\rm per}^\infty([0, 1])$ denote the space of infinitely continuously differentiable, periodic functions $[0, 1] \to \mathbb{R}$, i.e., $$ f(0) = f(1) = 0. $$ Let $P = \{z_1, \dots, z_n\} \subset [0, 1]$ and define $$ V(P) = \{u \in X : u |_P = 0\} = \bigcap_{z \in P} \{u \in X : u(z) = 0\}. $$ This is the set of functions in $X$ that vanish on $P$.
It seems that $V(P)$ is a subspace of codimension $n$ within $X$. What can be said about the structure of $V(P)$? Is it true that every subspace of $X$ with codimension $n$ can be written as $V(P_n)$ for some $|P_n| = n, P_n \subset[ 0, 1]$? I am particularly interested in the case where $P$ is comprised of $z_1, \dots, z_n$ which are independently, and uniformly distributed on $[0, 1]$.