I am using this document as a reference on tangent spaces etc. In the section on tangent spaces, the author provides three equivalent definitions of a tangent vector, the first being the intuitive equivalence class of curves and the second corresponding to a specialized instance of the Zariski tangent space for manifolds (i.e. germs of functions).
The third definition is as follows:
Definition (Tangent Vectors, version 3): Let $M$ be a $C^{k\geq 1}$ $n$-manifold and let $p\in M$. A tangent vector at $p$ is an equivalence class under the relation $\sim_p$ on the set of triples $(U,\varphi,u)$ where $(U,\varphi)$ is a chart at $p$ and $u\in\mathbb{R}^n$ and where $$(U,\varphi,u)\sim_p (V,\psi,v)
\iff \left(\psi \circ \varphi^{-1}\right) '_{\varphi(p)}(u)=v
\iff \lim_{x \to\varphi(p)} \frac{\lVert (\psi \circ \varphi^{-1})(x) – \psi(p) – v \rVert}{\lVert x – \varphi(p) \rVert}=0$$
We can then define a map $\theta_{U,\varphi,p}(u)=[(U,\varphi,u)]$ from $\mathbb{R}^n$ onto our newly-defined $T_{p}M$, which is clearly an isomorphism.
My question concerns the definition of $\sim_p$ in terms of the differential $(\psi\circ\varphi^{-1})'_{\varphi(p)}$; namely, how do we readily show that $\sim_p$ is an equivalence relation; i.e., show that $\sim_p$ is symmetric in that
$$\lim_{x \to\varphi(p)} \frac{\lVert (\psi \circ \varphi^{-1})(x) – \psi(p) – v \rVert}{\lVert x – \varphi(p) \rVert}=0
\iff \lim_{x\to\psi(p)} \frac{\lVert (\varphi\circ\psi^{-1})(p) – \varphi(p) – u \rVert}{\lVert x – \psi(p) \rVert} = 0$$
and similar for transitivity and reflexivity (I assume the proofs are in the same vein)? Unfortunately my knowledge of functional analysis and the Fréchet derivative is a bit limited.
Best Answer
I know this is an old post, but I might as well answer in case someone stumbles across it.
You don't need Frechet derivatives or functional analysis. In a finite-dimensional vector space, the Frechet derivative is just the regular multi-variable derivative, i.e. the linear map whose matrix is the Jacobian.
Since by the definition of a $C^k$ manifold each transition map $\psi \circ \phi^{-1}$ is inverse to the reverse transition map $\phi \circ \psi^{-1}$, the regular chain rule for multi-variable calculus implies their derivatives are inverses for each other. We are just talking about linear maps or matrices inverse to each other since this is finite-dimensional (the manifolds are locally diffeomorphic to $\mathbb{R}^n$ for finite $n$).
Once you know these derivatives are mutually inverse, it's simple to prove that the relation is symmetric without having to use the limit definition. The limits are handled in the proof of the chain rule.