# Infinite dimensional tangent spaces

Let $$p \in \mathbb{R}^n$$ . It is well-known that there are at least two equivalent definitions of the tangent space at $$p$$:

1. $$T_p \mathbb{R}^n$$ is the set of equivalence classes of $$C^1$$-curves $$\gamma : (-1,1) \to \mathbb{R}^n$$ with $$\gamma(0)=p$$ and where two curves are equivalent, iff they share the same derivative at $$0$$.

2. (The Zariski way) $$T_p \mathbb{R}^n$$ is the set of all derivations $$D : C^\infty(\mathbb{R}^n,\mathbb{R})_p \to \mathbb{R}$$ on the stalk at $$p$$ of the sheaf of smooth functions.

The trick to prove the equivalence of both definitions is the fact, that for every smooth function $$f: U \to \mathbb{R}$$, $$U \ni p$$ open, there are $$\varepsilon <0$$, smooth functions $$f_1, \dots , f_n$$ with $$f_i(p)= \frac{\partial f}{\partial x^i}(p)$$ and $$f(x^1, \dots , x^n) = \sum_{i=1}^n x^i \cdot f_i(x^1, \dots , x^n) \qquad (x^1, \dots, x^n) \in \mathbb{B}_\varepsilon(p)$$ The $$f_i$$ are defined via $$f_i : x \mapsto \int_0^1 \frac{\partial f}{\partial x^i}(p + t(x-p) \ dt$$ Note that, if $$f \in C^n(\mathbb{R}^n,\mathbb{R})_p$$, the functions $$f_i$$ are only $$C^{n-1}$$, so this trick would fail if being used for $$C^n$$-manifolds.

Now let $$E$$ be an arbitrary real Banach space and $$p \in E$$. As before, we have at least two options to define the tangent space $$T_pE$$. Every equivalence class of curves, sharing the same derivative, yields a derivation. But I see no proof for the converse. If $$f : U \to \mathbb{R}$$ is a smooth function on an open neighborhood of $$p$$, there exists an $$\varepsilon >0$$ and a smooth function $$A : \mathbb{B}_\varepsilon(p) \to E'$$, $$E'$$ being the continuous dual space of $$E$$, such that $$f(x) = \langle A(x), x \rangle \qquad x \in \mathbb{B}_\varepsilon(p)$$ and $$A(p)=df(p)$$ where $$df : E \to E'$$ is the Frechet-differential of $$f$$. $$A$$ is defined via a Bochner integral $$A :x \mapsto \int_0^1 df(p+t(x-p)) \ dt$$ At this point, the product rule for derivations cannot help us.

Note that the curve-definition leads to a tangent space isomorphic to $$E$$ while I guess the derivation space to be "larger".

My question: How to define $$T_pE$$? And how to do it, if $$E$$ is a complex Banach space?