I am looking for a quick definition of the tangent space $T_p M$, where $p$ is a point of a smooth manifold $M$. I mean a definition that allows easily to prove that: (1) $T_p M$ is a vector space of the same dimension of $M$; (2) if $x^1, x^2, \ldots,x^d$ are local coordinates functions then $\partial / \partial x^i|_p$ is a basis of $T_p M$; (3) The elements of $T_p M$ are the same thing of derivations of functions definited on a neighborhood of $p$ (W. Warner, Foundations of Differentiable Manifolds, called them "germs" and denoted their set by $\tilde{F}_p$).
Surely the old-style definition of $T_p M$ by equivalence classes of curves which passes through $p$ is inadequate.
The definition of $T_p M$ like the vector space of derivations of germs is, obviously, OK for the point (3) but seems to me that points (2) and (1) require to look at the cotangent space $T_p^* M$. I do not like to talk about the cotangent space before I have finished to talk about the tangent space.
I saw that someone defines the cotangent space first and then the tangent space, however the Warner definition of cotangent space like $\tilde{F}_p / \tilde{F}_p^2$, where $\tilde{F}_p^2$ is the ideal of finite linear combinations of product of two germs, I think it is rather artificial, and justifiable only in retrospect.
In conclusion I really appreciate any advice on how to define $T_p M$ so that (1), (2) and (3) are easy to prove. Thanks.
Best Answer
I think the description of tangent space $T_pM$ you're looking for is one defined entirely locally by "pushing forward" tangent spaces of $\mathbb{R}^n$ via local coordinate charts. ie. the tangent space $T_pM$ at a point $p$ in a trivial neighborhood $U$ of the manifold $M$ is described via a coordinate chart $\phi: (\mathbb{R}^n, 0) \to (U,x)$ as simply the image of $T_0 \mathbb{R}^n$ under the differential $d\phi$. This yields your (1), (2) immediately, while (3) follows once one understands why its true on $\mathbb{R}^n$.
The basic point of smooth manifolds, I think, is that there is no calculus on manifolds, but only calculus on $\mathbb{R}^n$ and pushforwards and pullbacks.
When I was first learning the basics of smooth manifolds I found Warner's book to be very helpful. He demonstrates very clearly the formalism of (co)tangent spaces and how to actually perform differential computations.