Let $(X,\Sigma,\mu)$ be a finite nonzero measure space, and let $L_0$ the set of (equivalence classes of) measurable real-valued functions on it.
This should be a classical question, but why $L_0$ cannot be normed?
Ps. Also a reference would be fine, since I imagine this is straighforward..
Edit, could we use this result (see here): A locally convex Hausdorff space is normable if and only if it has a bounded neighborhood of zero.