Let $H$ be the space of functions $\alpha: [0, T] \longrightarrow \mathbb{R}^n$ that are absolutely continuous and such that $\alpha(0)=0$. The statement that I have implicitly found in a paper is that this is a Hilbert space with the scalar product $ (\alpha, \beta) = \int_0^T \langle \dot{\alpha}(t), \dot{\beta}(t)\rangle \mathrm{d} t.$
The first question is why this is even well-defined. I know that the product of absolutely continuous functions is absolutely continuous and that an absolutely continuous function has a derivative in $L^1$, but in the integral, this is something like a second derivative.
The second question is about the completeness. Are there good references?