Do sections of a vector bundle over manifold form a Hilbert space?

Question

Let $\pi:E\rightarrow M$ denote a real (or complex) vector bundle of rank $r$ with a Riemannian (or Hermitean) fibre metric $\langle\cdot,\cdot\rangle$, over a smooth oriented manifold $M$ and let $\Gamma(E)$ denote the vector space of smooth sections of $E$. Fix a volume form $\omega$ of $M$, and for sections $f,g\in \Gamma(E)$ we set $$\langle f,g\rangle_{L^2}=\int_M \langle f, g\rangle\omega $$ then is the space $$L^2(E)=\{f\in\Gamma:\langle f,f\rangle_{L^2}<\infty\}$$ a Hilbert space with respect to $\langle\cdot, \cdot\rangle$? If not, what is an assumption for the $M$ for which it holds?