Let $R$ be a self-adjoint nonnegative definite ($n\times n$) matrix. Consider the class of $n\times 1 $ functions $u(\cdot)$, Lebesgue measurable on $(0,1)$, and such that $$ \| u \|^2 = \int_{0}^1[ R u(t), u(t) ] dt < \infty. $$ Can this be made into a Hilbert space with norm $\|u\|$?
Here $[\cdot,\cdot]$ is just an inner product between finite-dimensional vectors.
I think the approach is to show that given a Cauchy sequence in this function space, show that it converges, so it should use $$\|u_n - u_m\| < \epsilon $$ for all $n, m$ greater than $N(\epsilon)$. Expanding this would give $$\left[\int_{0}^1\overline{u_n}Ru_ndt \,\,-2\Re(\int_{0}^1\overline{u_m}Ru_ndt) + \int_{0}^1\overline{u_m}Ru_mdt \right]^{0.5} < \epsilon.$$ I'm not sure how to proceed in showing if this is a Hilbert space or not.
You have to assume that $R$ is strictly positive definite since you don't even get a norm in general. (e.g., when $R=0$). But in this case there exist positive constants $c$ and $C$ such that $c \|x\|^{2} \leq \langle Rx, x \rangle \leq C\|x\|^{2}$ for all $x$ . So your norm is equivalent to the $L^{2}$ norm from which completeness follows easily.