Given an open subset $U \subseteq \mathbb{R}^n = X$ the study of the $\mathbb{R}$-algebra $A = C^{\infty}(U, \mathbb{R})$ yelds the notion of derivations $\mathcal{D}^1(U)$ and differential forms.
In differential geometry it is very important to note that derivations
can be localwise seen (= tangent vectors) as classes of smooth curves.
For $p \in U$ one considers smooth curves
$$
v \colon (-\varepsilon, \varepsilon) \to U
\text{ such that }
v(0) = p
$$
modulo the relation $\dot{v}_1(0) = \dot{v}_2(0)$.
This defines the Gateaux derivations $\mathcal{GD}_p(U)$.
I think of this as a geometric object and not an algebraic one.
The relation with derivations is given as follows:
if $f \colon U \to Y$ is Frechet smooth at $p \in U$, for every
$\underline{v} \in X$ there is $\varepsilon > 0$ such that
$$
\begin{align}
v \colon (- \varepsilon, \varepsilon) &\to U \\
t &\mapsto p + t \, \underline{v}
\end{align}
$$
is contained in $U$ and one has
$$Df(p)(\underline{v}) = \partial_t (f \circ v)(0).$$
Any $v \in \mathcal{GD}_p(U)$ defines a $\mathbb{R}$-derivation at $p$ as $$ \begin{align} v \colon C^{\infty}_p(U, \mathbb{R}) &\to C^{\infty}_p(U, \mathbb{R}) \\ f &\mapsto [v(f)]. \end{align} $$
This should define a correspondence $$ \mathcal{GD}_p(U) \stackrel{1:1}{\longleftrightarrow} \mathcal{D}^1_p(U). $$
I don't understand this construction, hence the following questions.
- Is there a nice way to construct the Gateaux derivations?
One starts with a sheaf $\mathcal{G}$ on $\mathbb{R} = Y$ defined as $$\mathcal{G}(I) = C^{\infty}(I, U)$$ which does not have an algebraic structure. What kind of structure does it posses? Can we get to $\mathcal{GD}_p$ through some categorical constructions? - Can we expect to replace $X = \mathbb{R}^n$ and $Y = \mathbb{R}$ with Banach spaces?
How should $[v(f)]$ be defined? Is the following correct?
Given $v \in \mathcal{GD}_p(U)$, $\dot{v}(0) = \underline{v}$ and $f \colon B_r(p) \to \mathbb{R}$ then for $q \in B_r(p)$ one considers $$v_q(t) = q + t \underline{v}$$ over an open convex $I_q \ni 0$ such that $v_q$ is contained in $B_r(p)$. Then one sets $$ v(f)(q) = \lim_{t \rightarrow 0} \frac{1}{t} (f(v_q(t)) - f(v_q(0))$$ and then considers the germ of $v(f)$ at $p$, which hopefully is well defined.