Let G be a locally compact group and \({\mathcal L}(G)\) its Lie algebra in the sense of R. Lashof. We introduce the notion of a projective basis as a particular vector space basis of \({\mathcal L}(G)\) and prove the existence of such a basis for every locally compact group. From this we obtain results concerning the topological structure of \({\mathcal L}(G)\). Furthermore the existence of a projective basis allows to consider Banach spaces of k- times uniformly differentiable functions on G, containing the space \({\mathcal D}(G)\) of test functions as a dense subspace.