determinant condition for a sequence of vectors

Let $x_{1},x_{2},...$ be a sequence of $d$ dimensional vectors. Assume that there is a function $C\colon{\mathbb{N}}^{d}\to{\mathbb{R}}\smallsetminus\{0\}$ such that

\sum_{\begin{subarray}{c}n_{1}+\cdots+n_{d}=n\\ 0<n_{1}<\cdots<n_{d}\end{subarray}}C(n_{1},...,n_{d})\det[x_{n_{1}},x_{n_{2}},% ...,x_{n_{d}}]=0

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

Title	determinant condition for a sequence of vectors
Canonical name	DeterminantConditionForASequenceOfVectors
Date of creation	2013-03-22 14:33:44
Last modified on	2013-03-22 14:33:44
Owner	GeraW (6138)
Last modified by	GeraW (6138)
Numerical id	10
Author	GeraW (6138)
Entry type	Result
Classification	msc 15A15