Definition 7

Tag 05NP, permanent. Cite it as [Bourbaki, Tag 05NP].

Algebra, chapter II, § 2. Modules of linear mappings. Duality

If E is a free module with a finite basis $(e_t)$ , the basis $(e_t^*)$ of the dual E of E defined by relations (23) is called the dual basis of $(e_t)$ .

Relations (23) can also be written in the form

\langle e_t, e_{t'}^* \rangle = \delta_{tt'}

where

\delta_{tt'}

is the Kronecker symbol on

T \times T

Note that if $T$ is finite and $(e_t^*)$ is the dual basis of $(e_t)$ , then, for

x = \sum_{t \in T} \xi_t e_t \in E, \quad x^* = \sum_{t \in T} \xi_t^* e_t^* \in E^*,

\langle x, x^* \rangle = \sum_{t \in T} \xi_t \xi_t^*.

The dual basis of a finite basis of a right A-module is of course defined similarly.

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

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Cites

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