Isomorphism between the Tangent Space and Euclidean Space

Question

I attempt to understand the invariance of dimension of topological manifolds from Loring Tu's An Introduction to Manifolds (Second Edition, page no. 89). The corresponding section of the book is given below.

My Question

In the proof of Corollary 8.7, it is claimed that the tangent space $T_p(U)$ at point $p \in U \subset \mathbb{R}^n$ is isomorphic to the Euclidean $n$-space $\mathbb{R}^n$. How do we know that this is the case? In other words, what is the proof of this claim?

Answer 1

In Tu's book, it is considered proved at page 10, section 2.1, where $T_{p}\mathbb{R}^n$ is "defined" as the vector space of all arrows emanating from a point $p \in \mathbb{R}^n$.

Besides the fact there is a misprint in the book, (should be $T_{F(p)}V$ isomorphic to $\mathbb{R}^m$ in the last line) if you have shown it is an $n$-dimensional vector space, then this is equivalent to say it is isomorphic to $\mathbb{R}^n$. If otherwise, you have not proved this, then your line of proof is to build a linear injective and surjective map from $\mathbb{R}^n$ to $T_{p}\mathbb{R}^n$.

In general, for a smooth $n$-dimensional manifold, the tangent space at a point of the manifold will be a vector space isomorphic to $\mathbb{R}^n$. Proving this may be more or less difficult, depending on which of the many (mostly equivalent) definitions of manifold (and tangent space) you're using.

If you go on to Tu's chapter 8 section 4 you find "Bases for tangent spaces at a point". In this section, you prove formally that there is an isomorphism" in the general case of manifolds.

Answer 2

Let $A$ be an affine space with translation space $V$. That is, $A$ is a set whose elements are points, and we have an operation $+:A\times V \to A$ such that $p+0=p$ for all $p \in A$, we have $(p+v)+w=p+(v+w)$ (where the last $+$ is addition in $V$) and given $p,q \in A$, there is a unique $v\in V$ such that $q=p+v$ (people write $v = q-p$).

Once $O \in A$ has been chosen, we have a bijection $V \to A$ given by $v\mapsto O+v$, and so every structure $V$ has (vector space structure, topology, manifold structure, etc) is transferred to $A$. A different choice of $O$ gives an isomorphic vector space structure, same topology, and same manifold structure.

For every $p \in A$, we have $T_pA =V$ in a natural way. For example, if $$T_pA = \{\alpha'(0)\mid \alpha:(-\epsilon,\epsilon)\to A \mbox{ with }\alpha(0)=p\},$$one can write $$v = \frac{\rm d}{{\rm d}t}\bigg|_{t=0}p+tv.$$If you define $T_pA$ as the space of derivations of germs of smooth functions at $p$, then the isomorphism is $$v\mapsto \left([f]_p \mapsto \frac{\rm d}{{\rm d}t}\bigg|_{t=0}f(p+tv)\right),$$and so on.