I don't understand how author associate the smooth manifolds and linear subspace. TM is a linear subspace,what 's the mean of T?A set of vector?
And find the definition on Wikipedia.
I still don't understand what's tangent space,because I dont know what's the mean of Ck.
then I looked up it on wolfram mathworld.
I don't know what 's the mean of "attach at x a copy of R^n tangential to M." How to define a space tangential to a manifold?
If you have a manifold embedded in some $\mathbb R^n$, and a set of coordinates on that manifold $x^1, x^2, \ldots, x^m$, then each point on the manifold is a position vector, a function of the coordinates like so: $r = r(x^1, x^2, \ldots, x^m)$.
Each partial derivative of that function, $\partial r/\partial x^1, \partial r/\partial x^2, \ldots$ is a vector field on the manifold. At a given point $p$, they give us so-called a basis of tangent vectors--vectors tangent to the manifold.
The span of these tangent vectors at $p$ gives us the tangent space at $p$. Because each of these tangent spaces is a flat vector space, they are essentially the same as copies of $\mathbb R^m$.