Differing definitions of an inner product

[Kreizhn]

Hey all,

This might seem like a stupid question, and this might not be the correct forum, but hopefully someone can clarify it really easily.

I often have seen two definitions of an inner product on a vector space. Firstly, it can be defined as a bilinear map on a $\mathbb F-$vector space V as
[tex] \langle \cdot, \cdot \rangle : V \times V \to \mathbb F[/itex]
satisfying the usual inner product conditions. An example that comes to mind is the Riemannian metric, which is a 2-tensor and so acts on two copies of a tangent space. Alternatively, I've seen it defined as
$$\langle \cdot, \cdot \rangle : V^* \times V \to \mathbb F$$
satisfying the usual inner product conditions. An example that comes to mind here is the formalism used in the Riesz Representation theorem.

The only place I've really seen the first definition is in the case of Riemannian metrics, hence the motivation for posting this discussion in this forum.

Now I know that for finite dimensional vector spaces that V and $V^*$ are isomorphic: is this the reason for the differing notations? Or is it perhaps that in the second case when the domain is $V^*\times V$ we've some how canonically identified a vector $v \in V$ with its induced linear functional
$$v \mapsto \langle v, \cdot \rangle$$?

Again, this may seem really simple but I'd appreciate any response.

 

[arkajad]

The pairing

$$\langle \cdot, \cdot \rangle : V^* \times V \to \mathbb F$$

i s natural and it is not inducing an extra structure on V. On the other hand

$$\langle \cdot, \cdot \rangle : V \times V \to \mathbb F$$

is essentially the same as fixing a particular identification of V and V* (though it is somewhat more complicated for sesquilinear scalar products).

 

[Kreizhn]

Are those supposed to be the same domains? And is there a particular identification, like the one I mentioned in my original post?

 

[arkajad]

I have made an error in my comment (unnecessary star in the second formula). Now corrected.

 

[Hurkyl]

Alternatively, I've seen it defined as
$$\langle \cdot, \cdot \rangle : V^* \times V \to \mathbb F$$
satisfying the usual inner product conditions. An example that comes to mind here is the formalism used in the Riesz Representation theorem.

While I have seen the natural product on V^* x V written with that notation,
I have never seen it called an "inner product", except when being sloppy. The big point of the Riesz Representation theorem is that for the particular topological inner product space of interest, every continuous linear functional is the transpose of a vector.