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