Pairing of vector spaces

Question

Suppose we have a pairing i.e. a bilinear map $\phi : V \times V \rightarrow \mathbb{R} $. There are many ways of getting a map $ \psi: V \rightarrow V^* $ two of which are $v\rightarrow \langle v,.\rangle $ and $v\rightarrow \langle .,v\rangle $. Which of these should be called a canonical map associated with the pairing ? If the pairing is symmetric to start with, these two ways are the same but not in the general case. Is there a standard convention followed in the literature which associates to each pairing a linear transformation from the vector space to its dual ?

Answer 1

The only way I can think of breaking your tie is by looking to your choice of convention for a sesquilinear complex-valued pairing $\Phi : H \times H \to \mathbb{C}$ (e.g., the inner product on a complex vector space), since taking the real part $\Re \Phi : H \times H \to \mathbb{R}$ is a fairly common source of real-valued pairings, at least in analysis:

1. Most pure mathematicians adhere to the convention that $\Phi$ sesquilinear is linear in the first argument (i.e., $\Phi(av_1+v_2,w) = a\Phi(v_1,w)+\Phi(v_2,w)$) and antilinear in the second argument (i.e., $\Phi(w,av_1+v_2) = \bar{a}\Phi(w,v_1)+\Phi(w,v_2)$); given this, the canonical map $H \to H^\ast$ is forced to be the antilinear map $v \mapsto \Phi( \cdot,v )$. Hence, consistency with this convention for complex-valued sesquilinear pairings might break the tie in favour of $v \mapsto \phi(\cdot,v)$ in the real case.
2. Mathematical physicists, on the other hand, adhere to the convention that $\Phi$ sesquilinear is linear in the second argument and antilinear in the first; given this, the canonical map $H \to H^\ast$ is forced to be the antilinear map $v \mapsto \Phi(v,\cdot)$. Hence, consistency with this convention might break the tie in favour of $v \mapsto \phi(v,\cdot)$ in the real case.

Of course, as Hagen von Eitzen points out in his comment, in the case of your real pairing, you really do have two canonical maps, so any tiebreaker is going to be entirely artificial, as above.