Remark 2

Let $A(t) = (a_{ij}(t))$ be a square matrix of order $n$ whose entries are real (resp. complex) regulated functions of $t$ on $J$, and let $C(t, t_0) = (c_{ij}(t, t_0))$ be the resolvent matrix of the corresponding linear system (3) (IV, p. 22). Let F be an arbitrary complete normed space over $\mathbf{R}$ (resp. $\mathbf{C}$) and let us consider the system of linear differential equations

where the unknown functions $y_j$, take their values in F. It is immediate that the solution $(\mathbf{u}_j)_{1 \leq j \leq n}$ of this system such that $\mathbf{u}_j(t_0) = \mathbf{d}_j$ for $1 \leq j \leq n$ ($\mathbf{d}_j$ being arbitrary in F) is given by the formulae

We consider in particular the case where $A(t)$ is an endomorphism of a vector space E of finite dimension $n$ over $\mathbf{C}$, such that there exists a basis of E with respect to which the matrix of $A(t)$ has real elements for every $t \in J$. Then the above shows (by th. 1 of IV, p. 179) that the resolvent matrix $C(t, t_0)$ with respect to the same basis also has real elements: it suffices to consider the vector space $E_0$ over $\mathbf{R}$ generated by the basis of E under consideration, and to remark that the restriction of $A(t)$ to $E_0$ is an endomorphism of this vector space.