Proposition 5

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

Integration, IX, p. 81, § 6. Promeasures and measures on a locally convex space

Let E and E₁ be two locally convex spaces, and u a continuous linear mapping of E into E₁. Let Q be a positive quadratic form on E', and Q₁ the positive quadratic form Q $\circ {}^t u$ on E'₁. Then $u(\Gamma_Q) = \Gamma_{Q_1}$ .

Set $\mu = u(\Gamma_Q)$ . By formula (4) of No. 3,

\mathscr{F}\mu = (\mathscr{F}\Gamma_Q) \circ {}^t u = e^{-Q/2} \circ {}^t u = e^{-Q_1/2} = \mathscr{F}\Gamma_{Q_1} ,

whence $\mu = \Gamma_{Q_1}$ by Prop. 3 of No. 3.

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

Cited by

§ 6. Promeasures and measures on a locally convex space

Cites

Prop. 3