# fa.functional analysis – Extension of a Hilbert basis

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.