Remarks

1) Let $V_i$ ($1 \leq i \leq n$) be $n$ normed vector spaces over K, and $\varphi$ a function defined on $V_1 \times V_2 \times \cdots \times V_n$, with values in V; by passing to the quotient along $R_\infty$ the function $\varphi$ defines a map from

into $\mathscr{H}_\infty(\mathfrak{F}, V)$, which one most often denotes by $\varphi(\tilde{f}_1, \ldots, \tilde{f}_n)$ (Gen. Top., I, p. 67). For example, taking $\varphi$ to be the maps $(x, y) \mapsto x + y$ and $x \mapsto x\lambda$ ($\lambda \in K$), one thus defines, for any two germs $\tilde{f}, \tilde{g}$ in $\mathscr{H}_\infty(\mathfrak{F}, V)$, the elements $\tilde{f} + \tilde{g}$ and $\tilde{f}\lambda$, and one verifies immediately that the rules of composition $(\tilde{f}, \tilde{g}) \mapsto \tilde{f} + \tilde{g}$ and $(\lambda, \tilde{f}) \mapsto \tilde{f}\lambda$ define on $\mathscr{H}_\infty(\mathfrak{F}, V)$ a vector space structure over the field K; in this space $\tilde{O}$ is the class formed by functions equal to 0 on a set in $\mathfrak{F}$, and $-\tilde{f}$ is the class formed by functions equal to $-f$ on a set in $\mathfrak{F}$. In the same way, if V is an algebra over K, one defines on $\mathscr{H}_\infty(\mathfrak{F}, V)$ a second internal rule of composition $(\tilde{f}, \tilde{g}) \mapsto \tilde{f}\tilde{g}$ by taking $\varphi(x, y) = xy$; together with the two preceding rules this defines an algebra structure over K on $\mathscr{H}_\infty(\mathfrak{F}, V)$; if V has a unit element e then $\mathscr{H}_\infty(\mathfrak{F}, V)$ has as a unit element the class $\tilde{e}$ formed by the functions equal to e on some set in $\mathfrak{F}$; for $\tilde{f}$ to be invertible in $\mathscr{H}_\infty(\mathfrak{F}, V)$ it is necessary and sufficient that for some $f \in \tilde{f}$ there exists a $Z \in \mathfrak{F}$ such that $f(t)$ is invertible in V for every $t \in Z$ (in which case this condition is satisfied by every function in the class $\tilde{f}$).

2) With the same notation, let $\psi$ be a map of a subset of $\prod_{i=1}^n V_i$ into V; we denote by $\psi(f_1, f_2, \ldots, f_n)$ the function equal to $\psi(f_1(t), f_2(t), \ldots, f_n(t))$ at every point $t \in E$ where the $f_i(t)$ are defined and where the point $(f_i(t))$ belongs to the set where $\psi$ is defined $^1$. For example, $f + g$ is the function equal to $f(t) + g(t)$ at every point $t \in E$ where $f$ and $g$ are both defined. Observe that the map $(f, g) \mapsto f + g$ is not a group law on $\mathscr{H}(\mathfrak{F}, V)$, since if $f$ is not defined on all of E there is no function $g \in \mathscr{H}(\mathfrak{F}, V)$ such that $f + g = 0$.