Given a normed vector space $X$ (over $\mathbb R$ or $\mathbb C$) and distinct linearly independent vectors $x_1, x_2, \ldots$ such that $\sum_{i\ge 1} a_i\, x_i = 0$, is each $a_i = 0$?
If $x_i$'s are orthogonal (assuming $X$ is an inner-product space), then the answer is positive, but I am unsure of how to proceed for the above general case. Any lead?