determinant condition for a sequence of vectors

Let $x_1,x_2,\mathrm{\dots }$ be a sequence of $d$ dimensional vectors. Assume that there is a function $C:{\mathbb{N}}^d\to ℝ\setminus \{0\}$such that

for every $n\in \mathbb{N}$. Then the sequence is contained within a proper linear subspace.