Let's consider a set $U =\big\{f_{t} \big\}_{t \in \mathbb{R}}$. We define elements in $U$ as following: $f_t(x) = e^{tx}$.
Our task is to show that the set $U$ is linearly independent in $C \{[a, b] \}$, where $a, b \in \mathbb{R}$ and $C \{[a, b] \}$ means the space of continuous functions $g: [a, b] \rightarrow \mathbb{R}$.
I don't really know how to define linear independence if $t \in \mathbb{R}$.
I would appreciate any help or tips.
One approach is to use the fact that eigenvectors of distinct eigenvalues are linearly independent:
Define $\phi:C^1([a,b])\to C^1([a,b])$ by $f\mapsto f'.$ Then, note that $T=\left \{ t_1,\cdots,t_n \right \}$ is a set of $distinct$ eigenvalues of $\phi$ for $S=\left \{ e^{t_1x},\cdots,e^{t_nx} \right \}$ which says that $S$ is linearly independent.