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 $\mathbb{R}$?

For "the majority of practical uses" it is important that the Laplace transform ${\cal L}$ is injective. This means that when you have determined a function $s\mapsto F(s)$ that suits your needs, there is at most one process $t\mapsto f(t)$ such that $F$ is its Laplace transform. You can then look up this unique $f$ in a catalogue of Laplace transforms.
This injectivity of ${\cal L}$ is the content of Lerch's theorem and is in fact an essential pillar of the "Laplace doctrine". The theorem is proven first for special cases where we have an inversion formula, and then extended to the general case.
The difference between "injectivity" and "bijectivity" here is that we don't have a simple description of the space of all Laplace transforms $F$. But we don't need to know all animals when we want to analyze a zebra. Lerch's theorem tells us that it has a unique pair of parents.
