When dealing with a finite $d$-dimensional manifold $M$, one can define the tangent space $TM|_p$ of a manifold on a point $p \in M$ in different (but equivalent) ways, based on (at least) the following 3 ways to define a tangent vector:
- A tangent vector $v \in TM|_p$ is an equivalence class of curves $\gamma:\left ]-\varepsilon,\varepsilon \right[ \to M$ such that $\gamma(0) = p$ which have the same velocity $u \in \mathbb R^d$ on charts; that is, if $\phi: A \to \mathbb R^d$ is a chart around $p$, curves $\gamma_0$ and $\gamma_1$ are equivalent at $p$ when $(\phi \circ \gamma_0)'(0) = (\phi \circ \gamma_1)'(0)$.
- A tangent vector $v \in TM|_p$ is an equivalence class of pairs $(\phi,u)$, where $\phi$ is a chart and $u \in \mathbb R^d$, and $(\phi_0,u_0)$ and $(\phi_1,u_1)$ are equivalent at $p$ when $u_1 = D(\phi_1 \circ \phi_0^{-1})|_{\phi_0(p)}(u_0)$;
- A tangent vector $v \in TM|_p$ is a derivation at $p$: $v$ is a linear functional $v: \mathcal C^\infty(M,\mathbb R) \to \mathbb R$ such that, for every $f_0, f_1 \in \mathcal C^\infty(M, \mathbb R)$, $v(f_0 f_0) = v(f_0)f_1(p) + f_0(p)v(f_1)$.
Question Are these ways of defining $TM|_p$ equivalent if $M$ is an (infinite dimensional) Banach manifold modeled on a Banach space $E$?
Lang uses 2. to define the tangent space, and I believe (1) is equivalent to (2) in the infinite dimensional case (haven't done the calculations). I have tried to show that (2) and (3) are equivalent following the proofs on the finite dimensional case, but they all use the existence of a base. When trying to adapt the proofs, I found I could show the map from elements of (2) to elements of (3) is injective using Hahn-Banach, but to show it is surjective, I ended up having to use that the canonical morphism from the double dual $E^{**}$ to $E$ is surjective, which is not always true. Because of this, I believe (2) and (3) are inequivalent, but I am not sure if there is another way to do it.