Identification of each tangent space $T_pV$ with $V$ itself?

I found this statement from my text very confusing:

What does it mean by identification of each tangent space $T_pV$ with $V$ itself? - what does "identification" really mean here?

If it means isomorphic, then it conflicts with my understanding that each tangent space $T_pV$ is locally isomorphic to $V$.

What is $Xf$? I don't understand the expression. I don't know what is the function $f$.

Thank you very much for your help.

Solution 1:

What is implicit in the author's comment is the view of a tangent vector as a linear functional on the space of smooth functions defined near $p$.

This is a somewhat roundabout way of thinking of a tangent vector, and it does not connect easily with intuition, but it can be efficient when you study smooth manifolds.

The author's point is that when the manifold happens to be a vector space $V$, there is a natural choice for such a functional, namely the one given by the formula $Xf = \frac{d}{dt}\big\vert_{t=0}$, etc. In this case the roundabout approach is clearly redundant, but it needs to be shown that it is consistent with what we expect it to be intuitively, namely the directional derivative.

Solution 2:

One can think of the tangent spaces as the "directions one can move" (which include both speed and direction, i.e. they are vectors) on a manifold at a point $p$. It's not hard to convince yourself that if you look at a point on a vector space (which you should think of as flat in some sense), the directions you can move correspond to all the vectors in the space. If I start at a point of $\mathbb{R}^n$, I can move on $\mathbb{R}^n$ in any direction from $\mathbb{R}^n$. Not so if I start at a point on the two-sphere in $\mathbb{R}^3$, where I can move only in directions tangent to the two-sphere.

Now when you read "direction", you may replace it with "the operator on functions which is the directional derivative in that direction". Indeed, that is how geometers managed to make rigorous the notion of the directions you can move on a manifold at a given point.