Inequality with partial integration in one dimension Is it possible to prove $ \| u \|_{L^2(0,1)} \leq \| u' \|_{L^2(0,1)} $ for $u \in C^1([0,1])$ with $u(0)=0$ by using partial integration?
I don't know what partial integration is, but if you integrate 
$$|u(x)|^2 = \left|\int_0^x u'(t)\,dt \right|^2 \le \int_0^x |u'(t)|^2\,dt \le \|u'\|_{L^2}^2$$ 
over $0\le x\le 1$,  you get the desired inequality.
I don't know what partial integration is, but if you integrate 
$$|u(x)|^2 = \left|\int_0^x u'(t)\,dt \right|^2 \le \int_0^x |u'(t)|^2\,dt \le \|u'\|_{L^2}^2$$ 
over $0\le x\le 1$,  you get the desired inequality.