In his Linear Algebra, 4th ed. from 1975, Greub presents (p. 65) an abstract, symmetric definition of duality, in which two vector spaces $E^*,E$ over a field $\Gamma$ are said to be dual if there is a non-degenerate bilinear form $\langle-,-\rangle:E^*\times E\to\Gamma$ defined between them. This differs from most standard linear algebra books I've seen which simply define "the" dual space of a vector space $E$ to be the space $L(E)$ of linear functionals on $E$. Of course, with Greub's definition, there is an embedding $E^*\to L(E)$ defined by $x^*\mapsto\langle x^*,-\rangle$, which is an isomorphism when $E$ is finite-dimensional, but the definition itself is more general.
I prefer Greub's approach to the standard approach, and see it akin to many other uses of abstract definitions in mathematics -- like using the abstract definition of a group even though any group is isomorphic to a group of permutations (Cayley's Theorem), etc.
I am curious who first used this approach in linear algebra. I've found the following sources so far:
- Greub used it as early as his first German edition in 1958.
- In Fundamental Concepts of Algebra (1956), Chevalley uses it (p. 106) when defining "vector spaces in duality", although he also defines "the" dual module of a module earlier in the book (p. 67).
- In Lectures in Abstract Algebra, Vol. II: Linear Algebra (1953), Jacobson uses it (p. 141, 253).
I was expecting to find this approach used in Bourbaki, but I did not.
Does anyone have an earlier source?
