Exponential map from loop algebras to loop groups

Question

In Terry Gannon's book, 'Moonshine beyond the Monster', on page 206, he mentions that for a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, the exponential map from the loop algebra $L\mathfrak{g}$ to the loop group $LG$ is locally both one-to-one and onto. What exactly is meant by 'locally' here, and why is this statement true? Is it possible for the map to be globally onto, for special $G$? (I think it is important to note at this point that the loop group is an infinite dimensional Lie group).