I am trying to prove this statement from John M. Lee's book on smooth manifolds:
For a normed vector space $V$, the map $v\mapsto D_{v|a}$ such that $D_{v|a}(f)=\frac{d}{dt}\bigg|_{t=0}f(a+vt)$ is an isomorphism.
They say that we can argue like in the case of proving $\mathbb{R}^{n}\cong T_{a}\mathbb{R}^{n}$, where they show for a fixed point $a$ in $\mathbb{R}^{n}$ and a vector $v\in\mathbb{R}^n$, the map that takes $v$ to the directional derivative operator at $v$ is an isomorphism.
Following this I tried to prove the above:
Choose $\beta=\{v_1,v_2,...,v_n\}$, a basis of $V$.
Then fix a vector $v_{0}$.
For an arbitrary vector $v$, I will denote the coefficients of it's representation in terms of basis as $c_{i}$'s.
Let $\phi(v)=D_{v|v_0}$.
Then $\phi(v_{1}+v_{2})=D_{v_{1}+v_{2}|v_{0}}$.
For a smooth function $f\in C^{\infty}(V)$ we have
$$\lim_{t\to 0}\frac{f(v_{0}+(v_{1}+v_{2})t-f(v_{0})}{t}=\lim_{t\to 0} \frac{f(v_{0}+v_{1}t+v_{2}t)-f(v_{0}+v_{1}t)+f(v_{0}+v_{1}t)-f(v_{0}t)}{t}\\=\lim_{t\to 0}(D_{v_{2}+v_{1}t|v_{0}}+D_{v_{1}|v_{0}})(f)=(D_{v_{2}|v_{0}}+D_{v_{1}|v_{0}})(f).$$
I'll just omit the $D_{cv|v_{0}}$ part.
So this is linear.
If I denote the coordinate map $\pi_{i}:V\to\mathbb{R}$ wrt the basis $\beta$ such that $\pi(v)=c_{i}$.
Then using them we can get that if $v\in\ker(\phi)$. Then $D_{v|v_{0}}(\pi_{i})=c_{i}=0$ for $1\leq i\leq n$. So kernel is trivial.
However for proving the surjectivity of $\phi$ in the proof for $\mathbb{R}^{n}$, he uses Taylor's Theorem. Here's where I do not know how to proceed. I know that an injective linear map is a bijection for finite dimensional vector spaces but we have not proved the finite dimensionality of $T_{v_{0}}V$. The only way I can now try and think is that for a diffeomorphism $F$ from $V\to\mathbb{R}^{n}$, the spaces $T_{v_{0}}V\cong T_{F(v_{0})}\mathbb{R}^{n}$ are isomorphic and hence finite dimensional.
Is my thinking correct? Well, Lee says that the proof is similar. But as far as I see, there are more subtleties than I initially thought. Can anyone help me with this or maybe suggest an easier way. I am also not sure whether my proof of linearity of $\phi$ is correct.