Does the Laplace transform biject? Someone wrote on the Wikipedia article for the Laplace trasform that 'this transformation is essentially bijective for the majority of practical uses.'
Can someone provide a proof or counterexample that shows that the Laplace transform is not bijective over the domain of functions from $\mathbb{R}^+$ to $\mathbb{R}$?

Bijective from what space of functions to what space of functions?
For example, by the Paley-Wiener theorem the Laplace transform is a bijection from $L^2(0,\infty)$ to the functions $F$ analytic in the open right half plane whose restrictions to vertical lines in the right half plane have uniformly bounded $L^2$ norm.  
