Is the Laplace transform a vector space isomorphism? And what space is it isomorphic to?

Question

The laplace transform is a linear transformation, $\mathcal{L}: \mathcal{M} \rightarrow?$, where $\mathcal{M}$ is the set of exponentially bounded functions on $\mathbb{R},$since $\mathcal{L}(af(x)+bg(x))=a \mathcal{L}(f(x))+b\mathcal{g(x)}$ for $a,b\in \mathbb{R}$ and $f,g \in \mathcal{M}$. It seems to be injective since $\operatorname{Ker}(\mathcal{L})=0$ unless I've missed something. Therefore by the rank-nullity theorem $\mathcal{L}$ must surjective and so it is an isomorphism. So my questions are 1) Is this proof outline correct? and 2) what set is the laplace transform mapping into?