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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?

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