Remark 3

The principle of the preceding remark extends without significant modification to vector functions of a complex variable taking values in a complete normed space over $\mathbf{C}$. If $U$ is an open set in $\mathbf{C}$, homeomorphic to $\mathbf{C}$, a primitive of such a vector function $f$ defined on $U$ is by definition a continuous function on $U$, having derivative equal to $f$ at every point of $U$. With this definition, th. 1 of II, p. 52 extends without modification (one proves, using the connectedness of $U$, that $(g_\alpha)$ is uniformly convergent with respect to $\mathscr{F}$ on a neighbourhood of each point of $U$, from which it follows that $(g_\alpha)$ is uniformly convergent with respect to $\mathscr{F}$ on every compact subset of $U$; the proof is completed using prop. 4 of I, p. 18). Consequently, every function which is a uniform limit of polynomials on every compact subset of $U$, admits a primitive on $U$; these functions are no other than the functions called holomorphic on $U$, which we shall study further in detail in a later Book.