Studying an introduction to Hermitan inner products and complex spaces, I've found my self stuck to deal with a rather than classic example of an inner product.
The complete exercise goes as follows :
Let $ V =C([0,1])$, the complex space of the continuous functions : $[0,1] \to \mathbb C$.
Show that the following is an inner product :
$\langle x, y\rangle = \int_0^1f(x)\overline{g(x)}dx$
Now if V becomes the vector space of the bounded and partly continuous functions $\big($the bounded functions $f:[0,1] \to \mathbb C$ for which exists a finite sequence $ 0 = a_0 < a_1 < \dots < a_n = 0 $ with $f$ continuous in every $(a_i,a_{i+1})$ $\big)$ does the upper form continue to be an inner product ?
Do I need to show the properties of the Hermitan inner product ? If so, how do I proceed ? I don't even know where to start with this and it seems an elementary example since everywhere it's just used as an inner product and the proof is never asked for. I'd appreciate any help.
If you allow the functions in your space to be discontinuous at even one point, then you can consider the function $f$ that is $1$ at such a point, and $0$ everywhere else on the interval. Then $\int_0^1 |f|^2dt=0$ even though $f$ is not identically $0$. So that's the basic problem: you don't get positive definiteness.