I am proving that $C^0([a,b])$ is an infinite dimensional vector space. The fact that it is a vector space is clear. But I cannot understand how to prove that it has infinite dimension.
Let $F:=\{f_n:[a,b]\to\mathbb{R},x\mapsto x^n:n\in\mathbb{N}\}$. $\mid F\mid=+\infty$. But how can I prove that $F$ is made of linearly independent vectors?
Best Answer
Consider arbitrary finite collection of vectors $(x^{n_i})_{i\in\mathbb{N}_m}$ in $F$. Assume we have $$ \sum_{i=1}^m\alpha_i x^{n_i}=0 $$ This eqaulity tells us that we have a polynomial which is identically zero. Therefore all its coefficients are zero: $\alpha_i=0$ for all $i\in\mathbb{N}_m$. This means that $(x^{n_i})_{i\in\mathbb{N}_m}$ is linearly independent in $F$. Since we considered an arbitrary finite collection of vectors in $F$ and they turns out to be linearly independent, then the whole system $F$ is linearly idependent too.