Problem:
(a) Show that the $2\pi$-periodic function $u(x) = \lvert \sin \frac{x}{2} \rvert$ belongs to the periodic Sobolev space $H = H^1_{\text{per}}$, and is a weak solution of a problem of the form
$$-u''(x) + u(x) = f$$
where $f \in H^*$ is a linear combination of the Dirac delta distribution and a map of the form
$$v \mapsto \int_0^{2\pi}\overline{g(x)}v(x)dx$$
(b) Show that the function $w(x)=\sqrt{u(x)}$ does not belong to $H^1_{\text{per}}$. In particular, you must show that $w$ has no weak derivative in $L^2_{\text{per}}$. That is , there is no linear map $T: C_{\text{per}}^{\infty} \rightarrow \mathbb{C}$ that is bounded with respect to the $L^2$ norm such that
$$T(f) = -\int_0^{2\pi}\overline{w(x)}f'(x)dx \qquad \text{for all } f \in C_{\text{per}}^{\infty}$$
Question:
I know how to prove $u(x)$ belong to $H$ but I have no idea how to prove it is a weak solution of the ODE and (b). Actually, I'm lacking knowledge of finding or verifying a weak solution. Aside of solutions to this problem, I appreciate any general advice on verifiying weak solution of ODEs.