  This survey is about the fundamentals of the theory of finite dimensional Lie
groups over the field of real numbers. The notion of the tangent space of a
manifold at a point is considered to be defined via the well known chart and
vector formalism, here, a formalism equivalent to other commonly used ones
(namely, the curve and derivation methods). The proofs of all assertions about
Lie groups are in alignment with this formalism, here. Dealing with the basic
facts of Lie groups rigorously with regard to this formulation of tangent space
seems to be scarce in the literature.
