Let $0 \neq \phi \in L^2(\mathbb R^n)$ be a square-integrable function and $\mathcal F \subset \mathbb R^n$ a finite set. If we are in the one-dimensional setting $n=1$ then the set of translates of $\phi$ by $\mathcal F$, i.e. $$ \{ \phi(\cdot - x) : x \in \mathcal F \} \subset L^2(\mathbb R) $$ is linear independent. I found this result in the book Christensen, In introduction to frames and Riesz bases, p. 228.

I was wondering if this statement holds in arbitrary dimensions, i.e. without restriction of $n$ to $n=1$. If yes, does somebody knows a reference for this?

Thanks in advance!