Remark 3

Tag 0813, permanent. Cite it as [Bourbaki, Tag 0813].

— The definition (1) of the adjoint $u^*$ can also be written as

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

or, by V, p. 15, as

\langle u(x), y^*\rangle = \langle x, (u^*(y))^*\rangle .

But we also have

\langle u(x), y^*\rangle = \langle x, {}^t u(y^*)\rangle

, hence we can express the adjoint in terms of the transpose,

(u^*(y))^* = {}^t u(y^*) .

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

Cited by

Nothing.

Nothing.