The picture below is taken from this paper: http://real.mtak.hu/22877/.

The authors claim that the basis of $H^2(\Omega) \cap H^1(\Omega)$ denoted by $\lbrace w_i \rbrace _{i \geq 1}$ can be extended to be a basis of $L^2(\Omega;H^1(0,1))$. I don't see how it can be possible. In my thinking, we have to multiply $w_i(x)$ by $h_i(x)$ where $h_i(x)=\frac{cos(i \pi x )}{i\pi}$ is a basis of $H^1(0,1)$. Is this write?. Thank you.