Corollary

Tag 09SQ, permanent. Cite it as [Bourbaki, Tag 09SQ].

Integration, chapter IX, § 6. Promeasures and measures on a locally convex space

— Equip $E'$ with a topology compatible with its vector space structure. For $\Phi$ to be continuous, it is necessary and sufficient that its real part $\Re \Phi$ be continuous at the origin, in which case $\Phi$ is uniformly continuous.

This follows from the inequality (54).

Let F be a locally convex space. We equip the dual $F'$ of F with a topology compatible with the duality between F and $F'$ , and we identify F with the dual of $F'$ . Consequently, the Fourier transform of a bounded measure $\mu$ on $F'$ is the function $\mathscr{F}\mu$ on F defined by

(\mathscr{F}\mu)(x) = \int_{F'} e^{i\langle x, x' \rangle} d\mu(x').

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

Cited by

Proposition 9, § 6

Cites

Nothing.