The theorem says that if $U$ is an open subset of a real Banach space $E$ and $f, g_1, \dots , g_p : U \to \mathbb R$, $X = \{x \in U : g_1(x)= \dots = g_p(x)=0\}$, and $a \in X$ is such that $f_{|X}$ has a minimum at $a$ and $dg_1(a), \dots, dg_p(a) $ are linearly independent, then there exists $\lambda_1, \dots, \lambda_p \in \mathbb R$ such that $df(a)= \lambda_1 dg_1(a) + \dots + \lambda_p dg_p(a)$.
What happens if $E$ is not a Banach space ? Surely there are counter examples but I've tried some in $\mathbb R[X]$ and couldn't find one.