The proof is immediate.

By prop. 1, $\mathbf{u}(t, t_0, \mathbf{x}_1) + \mathbf{u}(t, t_0, \mathbf{x}_2)$ (resp. $\lambda \mathbf{u}(t, t_0, \mathbf{x}_0)$) is an integral of (4) and takes the value $\mathbf{x}_1 + \mathbf{x}_2$ (resp. $\lambda \mathbf{x}_0$) at $t_0$, so, by th. 1 of IV, p. 179, is identical to $\mathbf{u}(t, t_0, \mathbf{x}_1 + \mathbf{x}_2)$ (resp. $\mathbf{u}(t, t_0, \lambda \mathbf{x}_0)$); the map $\mathbf{x}_0 \mapsto \mathbf{u}(t, t_0, \mathbf{x}_0)$ is thus a linear map $C(t, t_0)$ of E into itself, and one can write $\mathbf{u}(t, t_0, \mathbf{x}_0) = C(t, t_0). \mathbf{x}_0$.

Since the map $(X, Y) \mapsto XY$ of $\mathscr{L}(E) \times \mathscr{L}(E)$ into $\mathscr{L}(E)$ is continuous (Gen. Top., X, p. 298, prop. 8), the map $t \mapsto A(t)U$ of J into $\mathscr{L}(E)$ is regulated for all $U \in \mathscr{L}(E)$; further (Gen. Top., X, p. 296)

so one can apply th. I of IV, p. 179 to the homogeneous linear equation (5); let $V(t)$ be the integral of this equation defined on J and equal to $I$ at $t_0$. One has (I, p. 6, prop. 3)

and for $t = t_0$ we have $V(t)\mathbf{x}_0 = I\mathbf{x}_0 = \mathbf{x}_0$; by th. 1 of IV, p. 179 one must have $V(t).\mathbf{x}_0 = C(t, t_0)\mathbf{x}_0$ for all $\mathbf{x}_0 \in E$, that is, $V(t) = C(t, t_0)$; this proves that $C(t, t_0)$ belongs to $\mathscr{L}(E)$, in other words, that $\mathbf{x}_0 \mapsto C(t, t_0).\mathbf{x}_0$ is continuous on E, and that the map $t \mapsto C(t, t_0)$ is the integral of (5) which is equal to $I$ at $t_0$.

Finally, the integral $s \mapsto C(s, u).\mathbf{x}_0$ of (4) is equal to $C(t, u).\mathbf{x}_0$ at the point $t$, so, by definition,

for any $\mathbf{x}_0 \in E$, whence the first relation (6); since $C(s, s) = I$ one has $C(s, t)C(t, s) = I$, for any $s$ and $t$ in J; this proves (Set Theory, II, p. 86, corollary) that $C(t, t_0)$ is a bijective map of E onto itself, with inverse map $C(t_0, t)$. This completes the proof of the theorem.

One says that $C(t, t_0)$ is the resolvent of equation (2) of IV, p. 178.

It is certainly a bijective linear map of E onto $\mathscr{I}$: now $C(t, t_0)$ is bounded on a compact set $K \subset J$, so $\| C(t, t_0).\mathbf{x}_0 \| \leq M \| \mathbf{x}_0 \|$ for any $t \in K$ and $\mathbf{x}_0 \in E$, which shows that this map is continuous; and since

it is clear that the inverse map is also continuous.

By (6) we have $C(s, t) = C(s, t_0) \left( C(t, t_0) \right)^{-1}$; now, the map $(X, Y) \mapsto XY$ of $\mathscr{L}(E) \times \mathscr{L}(E)$ into $\mathscr{L}(E)$ is continuous, as is the map $X \mapsto X^{-1}$ of the (open) group of invertible elements of $\mathscr{L}(E)$ onto itself (TG, IX, p. 40, prop. 14).

One may note that the map

admits a derivative equal to $-(C(t, t_0))^{-1} (dC(t, t_0)/dt)(C(t, t_0))^{-1}$ (on the complement of a countable set) (I, p. 8, prop. 4), that is to say (by IV, p. 179, formula (5)) equal to $-C(t_0, t) A(t)$.

Indeed, $\|A(t)x_0\| \leq k \|x_0\|$ for all $t \in K$; on K the constant function equal to $x_0$ is thus an approximate integral to within $k \|x_0\|$ by equation (4) of IV, p. 18; by formula (15) of IV, p. 170, one thus has

for any $t$ and $t_0$ in K, and $x_0$ in E, which is equivalent to the inequality (7) by the definition of the norm on $\mathscr{L}(E)$.

Indeed, by (5)

so $\frac{d}{dt}(BC - CB) = A(BC - CB)$; but $BC(t_0, t_0) - C(t_0, t_0)B = 0$, so (IV, p. 179, th. 1) $BC(t, t_0) - C(t, t_0)B = 0$ for all $t \in J$.

An important instance of prop. 2 is that where E is endowed with the structure of a normed vector space with respect to the field of complex numbers $\mathbf{C}$, and where, for every $t \in J$, $A(t)$ is an endomorphism of E for this vector space structure; this means that $A(t)$ commutes with the continuous endomorphism $x \mapsto tx$ of E (for the vector space structure over $\mathbf{R}$); then $C(t, t_0)$ commutes with this endomorphism, which means that for any $t$ and $t_0$ in J, the map $C(t, t_0)$ is a continuous endomorphism for the normed vector space structure of E over $\mathbf{C}$.

The method which leads to formula (10), which consists of taking the function $\mathbf{z}$ as a new unknown function, is often called the "method of variation of parameters".

Indeed, the map which to every $\mathbf{x}_0 \in E$ associates the integral $t \mapsto C(t, t_0).\mathbf{x}_0$ is an isomorphism of $E$ onto $\mathscr{I}$ (IV, p. 180, cor. 1 and IV, p. 181, prop. 2).

If $(\mathbf{e}_j)_{1 \leq j \leq n}$ is any basis of $E$ over $\mathbf{C}$, the $n$ integrals

thus form a fundamental system; if one identifies $C(t, t_0)$ with its matrix with respect to the basis $(\mathbf{e}_j)$ the integrals $\mathbf{u}_j$ are precisely the columns of the matrix $C(t, t_0)$. The integral of (4) that takes the value $\mathbf{x}_0 = \sum_{j=1}^n \lambda_j \mathbf{e}_j$ at the point $t_0$ is then $C(t, t_0).\mathbf{x}_0 = \sum_{k=1}^n \lambda_k \mathbf{u}_k(t)$.

Given any $n$ integrals $\mathbf{u}_j$ ($1 \leq j \leq n$) of (4) one terms the determinant of these $n$ integrals at a point $t \in J$ with respect to a basis $(\mathbf{e}_j)_{1 \leq j \leq n}$ of $E$, the determinant

of the $n$ vectors $\mathbf{u}_j(t)$ with respect to the basis $(\mathbf{e}_j)$ (Alg., III, p. 522). One has (Alg., III, p. 523, prop. 2)

By prop. 4 of IV, p. 184, for $(\mathbf{u}_j)_{1 \leq j \leq n}$ to be a fundamental system of integrals of (4) it is necessary and sufficient that the determinant $\Delta(t)$ of the $\mathbf{u}_j$ be $\neq 0$ at some one point $t_0$ of $J$; the formula (14) then shows that $\Delta(t) \neq 0$ at every point of $J$, in other words, that the vectors $\mathbf{u}_j(t)$ ($1 \leq j \leq n$) are always linearly independent.

Indeed, if one puts $\delta(t) = \det \left( C(t, t_0) \right)$ one has, by the formula for the derivative of a determinant (I, p. 8, formula (3))

that is, by the differential equation (5) of IV, p. 179 satisfied by $C(t, t_0)$,

Since $\delta(t_0) = 1$ the formula (15) follows from the expression (12) (IV, p. 183) for the integral of a scalar linear equation.

Specifying $n$ linearly independent integrals of (4) determines all the integrals of this equation, as we have just seen. We shall now show that for $1 \leq p \leq n$ the knowledge of $p$ linearly independent integrals $\mathbf{u}_j$ ($1 \leq j \leq p$) of equation (4) reduces the integration of this equation to that of a homogeneous linear system of $n - p$ scalar equations. Suppose that on an interval $K \subset J$ there are $n - p$ maps

of $K$ into $E$, which are primitives of regulated functions on $K$, and such that, for every $t \in K$, the $n$ vectors $\mathbf{u}_j(t)$ ($1 \leq j \leq n$) form a basis of $E$.

For every point $t_1 \in J$ there always exists an interval $K$, a neighbourhood of $t_1$ in $J$, on which there are defined $n - p$ functions $\mathbf{u}_{p+k}$ ($1 \leq k \leq n - p$) having the preceding properties. For, let $(\mathbf{e}_i)_{1 \leq i \leq n}$ be a basis of $E$; there exist $n - p$ vectors of this basis which form with the $\mathbf{u}_j(t_1)$ ($1 \leq j \leq p$) a basis of $E$ (Alg., II, p. 292, th. 2); suppose for example that they are $\mathbf{e}_{p+1}, \ldots, \mathbf{e}_n$; since the determinant $\det(\mathbf{u}_1(t), \ldots, \mathbf{u}_p(t), \mathbf{e}_{p+1}, \ldots, \mathbf{e}_n)$ (with respect to the basis $(\mathbf{e}_i)$) is a continuous function of $t$ and does not vanish for $t = t_1$ there exists a neighbourhood $K$ of $t_1$ on which it does not vanish; one can then take $\mathbf{u}_{p+k}(t) = \mathbf{e}_{p+k}$ ($1 \leq k \leq n - p$) for $t \in K$.

There exists an invertible matrix $B(t)$ of order $n$, whose elements are primitives of regulated functions on $K$, such that $B(t).\mathbf{e}_j = \mathbf{u}_j(t)$ for $1 \leq j \leq n$. Let us put $x = B(t).y$; then $y$ satisfies the equation $\frac{dB}{dt}.y + B(t).\frac{dy}{dt} = A(t)B(t).y$, which can also be written

where $H(t) = (h_{jk}(t))$ is a matrix with regulated entries on $K$. By the definition of $B(t)$ this linear equation admits the $p$ constant vectors $\mathbf{e}_j$ ($1 \leq j \leq p$) as integrals; one concludes immediately that necessarily $h_{jk}(t) = 0$ for $1 \leq k \leq p$; thus the components $y_k$ of $y$ (with respect to the basis $(\mathbf{e}_i)$) with index $k \geq p + 1$ satisfy a homogeneous linear system of $n - p$ equations; once the solutions of this system are determined, the $dy_j/dt$ for indices $j \leq p$ are linear functions of the $y_k$ with $k \geq p + 1$, so are known, and the primitives of these functions will give the $y_j$ for indices $j \leq p$.

In particular, when one knows $n - 1$ linearly independent integrals of equation (4) of IV, p. 183, integrating this equation reduces to that of a single homogeneous scalar equation, and then the evaluation of $n$ primitives.

where $-B(t)$ is the transpose of $A(t)$.

Equation (16) is called the adjoint of (4); clearly (4) is the adjoint of (16). The elements of the matrix $B(t)$ being regulated functions of $t$ on J, the results obtained above on linear equations apply to (16). In particular, the integral of (16) taking the value $\mathbf{x}_0^*$ at the point $t_0$ can be written as $H(t, t_0).\mathbf{x}_0^*$, where $H(t, t_0)$ is a bijective linear map of $\mathrm{E}^*$ onto itself, identical to the integral of the equation

which takes the value $I$ at the point $t_0$. As a result one has (with the notation of IV, p. 179)

for any $\mathbf{x}_0 \in \mathrm{E}$ and $\mathbf{x}_0^* \in \mathrm{E}^*$, which shows that

(the contragradient of $C(t, t_0)$). In particular, if one knows a fundamental system of integrals of the adjoint equation (16) the matrix $H(t, t_0)$ is determined, so is $C(t, t_0)$ also, and as a result, so are all the integrals of equation (4).