Rigorous definition of tangent space

Question

I am studying manifolds, while my book offers a definition of tangent space I am having problems understanding it, so what is a rigorous definition of tangent space?

Answer 1

There are several ways to define the tangent space. Here's one:

As the set of all derivations. A derivation at a point $p$ in a manifold $M$ is a linear map $X:C^{\infty}(M)\to\mathbb{R}$ which satisfies the product rule $$X(fg)=f(p)(Xg)+(Xf)g(p)$$ for all functions $f,g\in C^{\infty}(M)$. Then the tangent space $T_{p}M$ is the vector space of all derivations at $p$.

One can also define the tangent space in local coordinates, or via initial tangent vectors. Another good reference to look at is John M. Lee's Introduction to Smooth Manifolds, 2nd Edition. Chapter 3 in this book covers several definitions of the tangent space.