So, that's an exert from Hochschild's "The structure of Lie Groups" p.79. Existence and uniqueness of one-parametric Lie-Group embedding such that the given tangent vector of $G$ is the image of the unit canonical vector of R is being proved.
I don't understand the first sentence of the proof.
First of all, as far as I understand, $A$ is an open subset of $\mathbb{C}^n$, $U$ a subset of $G$ containing $1$. There is kind of action over functions on $G$, that is constructed by a tangent vector.
What's the initial and target spaces of $\gamma_U$? How does $\gamma_U$ constructed? Why does the following condition can be satisfied?
Hochschild is saying the following here:
Since $G$ is a real-analytic Lie group, we can identify (real-analytically) a neighborhood $A$ of $1\in G$ with an open subset $U\subset R^n$. Accordingly, the element $\gamma$ of the Lie algebra (regarded as a left-invariant real analytic vector field on $G$) becomes a real-analytic vector field on $U$, i.e. in the Cartesian coordinates $x=(x_1,...,x_n)$ on $R^n$, the vector field $\gamma$ can be written as $$ (p_1(x),...,p_n(x)), $$
where each real-analytic function $p_i$ can be represented as a convergent power series on $U$. (For the latter, one may have to shrink the neighborhood $A$.) All he needs at this point is to quote the Cauchy-Kovalevskaya theorem on differential equations with real-analytic coefficients. Maybe he does so on the next page...