My Doubt:-
I understood the proof of $T_p(\mathbb R^n)\simeq \mathcal{D}_p(\mathbb R^n)$. $T_p(\mathbb R^n)$ is a space consists of elements from $\mathbb R^n$. $\mathcal{D}_p(\mathbb R^n)$ is a space of all derivations at $p$. How it is possible to use equality between the quantity in equation (2.4)? I understood this $T(v)=\sum v^i\frac{\partial }{\partial x^i}|_p$. Can you please help me?
I was going to write this as a comment, but it became a tad too long.
People write this way mostly for convenience sake. Technically, you're absolutely right. As commentors have also pointed out, $T(v)$ is the correct thing to write. However, this kind of abuse of notation is fairly common in differential geometry. For example, "identification" is also used when writing points on the manifold using coordinates. You may see a statements like:
"Consider the point $ p =(a_1,\dots a_n)$ Given the chart $(U,x)$",
instead of the longer but more accurate
"Consider the point $p$ on the manifold with coordinates $(a_1,\dots a_n)$, or Let $p$ be so that $x(p) = (a_1,\dots a_n)$, with the coordinate chart $(U,x)$".
This instance is somewhat is similar. Further, for a manifold embedded in $\mathbb{R}^n$, we sometimes want to think about the tangent space as a subspace of $\mathbb{R}^n$. It's easy to write tangent vectors in whatever manner is more suitable for the purpose without having to bring in the $T$ or $T^{-1}$ again and again.
Differential geometry is abhorred by many students starting out (myself included) for it's unwieldy notation. After a point one gets used to it, and then the subject seems far more attactive. There's a running joke about differential geometry being a study of properties that are invariant under change of notation.
Here's a link to a reddit page on confusing Differential Geometry notation
Warning: It could also be a typo in the book, but I've seen this kind of thing often enough that it's worth a mention.