Definition 1

Tag 08D4, permanent. Cite it as [Bourbaki, Tag 08D4].

Functions of a Real Variable, chapter IV, § 2. LINEAR DIFFERENTIAL EQUATIONS

Given a continuous endomorphism A of E we denote by $e^A$ or $\exp A$ the automorphism of E equal to the value at the point $t = 1$ of the integral of the equation (20) which takes the value I at the point $t = 0$ .

With this notation the remarks preceding def. 1 show that

C(t, t_0) = \exp \left( A(t - t_0) \right).

The exponential notation just introduced is justified by the following properties, which are entirely analogous to those of the function exp z, for z real or complex (cf. III, p. 98 and 106):

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

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Cites

III, p. 98