1. ## Inner Product

Why do we define the inner product in the infinite-dimensional vector space $\displaystyle C([a,b], \mathbb{R})$ as $\displaystyle \langle f,g \rangle = \int_{a}^{b} f(x)g(x) \ dx$?

2. Originally Posted by Sampras
Why do we define the inner product in the infinite-dimensional vector space $\displaystyle C([a,b], \mathbb{R})$ as $\displaystyle \langle f,g \rangle = \int_{a}^{b} f(x)g(x) \ dx$?
It is not the inner product but a inner product on this space. However it is commonly seen as it gives us the $\displaystyle L_2$ norm, and is associated with many orthogonal function expansions, so is useful.

CB

3. It is only because in the case you are considering it's considered as a (non closed) subspace of $\displaystyle L^2([a,b])$. Endowed with its natural norm topology, $\displaystyle C([a,b],\mathbb{R}$ is not a Hilbert space.

4. It seems that Captain Black arrived a little bit earlier! Sorry because of the repetition!

5. Originally Posted by CaptainBlack
It is not the inner product but a inner product on this space. However it is commonly seen as it gives us the $\displaystyle L_2$ norm, and is associated with many orthogonal function expansions, so is useful.

CB
I wonder what would happen if we removed the condition of symmetry. In other words, we studied spaces that only had (i) bilinearity and (ii) positive definiteness.

6. it turns out that we do study inner products which are not symmetric. but we do not study arbitrary such products. for complex vector spaces we usually take conjugate symmetric inner products. i have not known of any other types as i guess it does not give us a rich enough structure to be useful