# Ngô Quốc Anh

## February 23, 2017

### In a normed space, finite linearly independent systems are stable under small perturbations

Filed under: Giải tích 8 (MA5206) — Ngô Quốc Anh @ 23:21

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 $(X, \|\cdot\|)$ is a normed space and $\{x_1,...,x_n\}$ is a set of linearly independent elements in $X$. Then $\{x_1,...,x_n\}$ is stable under a small perturbation in the sense that there exists some small number $\varepsilon>0$ such that for any $\|y_i\| < \varepsilon$ with $1 \leqslant i \leqslant n$, the all elements of $\{x_1+y_1,...,x_n+y_n\}$ are also linearly independent.

We prove this result by way of contradiction. Indeed, for any $\varepsilon>0$, there exist $n$ elements $y_i \in X$ with $\|y_i\| < \varepsilon$ such that all elements of $\{x_1+y_1,...,x_n+y_n\}$ are linearly dependent, that is, there exist real numbers $\alpha_i$ with $1 \leqslant i \leqslant n$ such that

$\displaystyle \alpha_1 (x_1+y_1) + \cdots + \alpha_n (x_n+y_n) =0$

with

$\displaystyle |\alpha_1| + \cdots + |\alpha_n| >0.$