Definition 1

Tag 080Y, permanent. Cite it as [Bourbaki, Tag 080Y].

Topological Vector Spaces, chapter V, § 4. Some classes of operators in hilbertian spaces

— Let E and F be two hilbertian spaces. For every continuous linear mapping $u : E \to F$ , the continuous linear mapping from F into E defined by formula (1) is called the adjoint of u and is denoted by $u^*$ .

We have

\begin{align*} (u + v)^* &= u^* + v^* \\ (\lambda u)^* &= \overline{\lambda} u^* \\ (u^*)^* &= u \\ (1_E)^* &= 1_E \\ (wu)^* &= u^* w^* ; \end{align*}

in all these formulas, $u$ and $v$ belong to $\mathscr{L}(E; F)$ , $\lambda$ is in K, and $w$ in $\mathscr{L}(F; G)$ where G is a hilbertian space. Formulas (5) and (6) mean that $u \mapsto u^*$ is semi-linear. Formula (8) is obvious. To prove (7), we take the conjugate of the two members of (1), which gives $\langle u^*(y)|x\rangle = \langle y|u(x)\rangle$ , and this proves that $u$ is the adjoint of $u^*$ . Finally, with the notations of (9), we have, for all $z \in G$

\langle w(u(x))|z\rangle = \langle u(x)|w^*(z)\rangle = \langle x|u^*(w^*(z))\rangle ,

hence $u^*w^*$ is the adjoint of $wu$ .

Let $u : E \to F$ be a bijective and continuous linear mapping; then it is also bicontinuous (I, p. 19, cor. 1). From (8) and (9) we immediately deduce that $u^*$ is bijective and bicontinuous and that

(10)

(u^{-1})^* = (u^*)^{-1} .

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

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