In this topic, we show that in a normed space, any finite linearly independent system is stable under small perturbations. To be exact, here is the statement.

Suppose is a normed space and is a set of linearly independent elements in . Then is stable under a small perturbation in the sense that there exists some small number such that for any with , the all elements of are also linearly independent.

We prove this result by way of contradiction. Indeed, for any , there exist elements with such that all elements of are linearly dependent, that is, there exist real numbers with such that

with