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

)}{ipi}$ is a basis of $H^1(0,1)$. Is this write?. Thank you.