Tangent space

This gives a degenerate norm on the tangent space of Hₙ at the origin, and one can use left translations to define similar norms at the tangent spaces of Hₙ at all other points.

In a Lie group G, because left translation by an element g #92;in G is a diffeomorphism that maps a neighborhood of the identity to a neighborhood of g, all the local information about the group is concentrated in a neighborhood of the identity, and the tangent space of the identity assumes a special importance. Moreover, one can give the tangent space TₑG a Lie bracket [ , ], so that in addition to being a vector space, it becomes a Lie algebra, called the Lie algebra of the Lie group.

More dramatically, there are situations where one has a sequence of smooth manifolds such that the limit is again a smooth manifold, but with a limiting “tangent space” which has nothing to do with the actual tangent space of the limit!