Proposition 2

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

Integration, chapter III, § 3. Integrals of continuous vector-valued functions

— Let $u$ be a continuous linear mapping of $E$ into a locally convex space $F$ ; for every function $f \in \widetilde{\mathscr{K}}(X;E)$ , the function $u \circ f$ belongs to $\widetilde{\mathscr{K}}(X;F)$ and

\int u(f(x))\, d\mu(x) = ^{tt} u \left( \int f(x)\, d\mu(x) \right).

For every continuous linear form $z' \in F'$ , we have $z' \circ u \circ f = y' \circ f$ on setting $y' = z' \circ u = {}^t u(z') \in E'$ , whence the first assertion; moreover, for all $z' \in F'$ ,

\left\langle \int (u \circ f)\, d\mu, z' \right\rangle = \int \langle u \circ f, z' \rangle\, d\mu = \int \langle f, {}^t u(z') \rangle\, d\mu \\ = \left\langle \int f\, d\mu, {}^t u(z') \right\rangle = \left\langle {}^{tt} u \left( \int f\, d\mu \right), z' \right\rangle,

whence the formula (2).

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

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