Notation: $H_0^1=H_0^1(a,b)$, where $-\infty<a<b<\infty$.
Let $\|\cdot\|_V$ be a norm on $V:=H_0^1\times H^1_0\times H^1_0$ given by $$\|(f,g,h)\|_V=\|f'+g+ah\|_{L^2}+\|h'-af\|_{L^2}+\|g'\|_{L^2},$$ where $a$ is a positive constant.
I need help to prove that $(V,\|\cdot\|_V)$ is complete. I have tried to show that $\|\cdot\|_V$ is equivalent to $|\cdot|_V$, where $|\cdot|_V$ is a norm such that $(V,|\cdot|_V)$ is complete, as explained below.
We know that the space $H_0^1$ is complete with the norm $\|f\|_{H^1}=\|f\|_{L^2}+\|f'\|_{L^2}$. Therfore $V$ is complete with the norm $$\begin{align*}|(f,g,h)|_V&=\|f\|_{H^1}+\|g\|_{H^1}+\|h\|_{H^1}\\ &=\|f\|_{L^2}+\|f'\|_{L^2}+\|g\|_{L^2}+\|g'\|_{L^2}+\|h\|_{L^2}+\|h'\|_{L^2} \end{align*}$$
By triangular inequality,
$$\|(f,g,h)\|_V\leq\|f'\|_{L^2} + \|g\|_{L^2} + a\|h\|_{L^2} + \|h'\|_{L^2} + a\|f\|_{L^2} + \|g'\|_{L^2}\leq\max\{1,a\}|(f,g,h)|_V$$
So, there exists $c:=\max\{1,a\}>0$ such that $\|(f,g,h)\|_V\leq c|(f,g,h)|_V$ for all $(f,g,h)\in V$.
Now, I need help to show that there exists a constant $d>0$ such that $|(f,g,h)|_V\leq d\|(f,g,h)\|_V$ for all $(f,g,h)\in V$.
Thanks.
I think it might not even be true it's equivalent to a norm like that.
My approach would be to show if $(f_n,g_n,h_n)$ is Cauchy in this norm, then $(f'_n + g_n+a h_n)$ , $(h'_n-af_n)$ and $(g'_n)$ converge to some $L^2$ functions, by completeness of $L^2$. Now you should be able to write all 6 of $f_n,g_n,h_n,f'_n,g'_n,h'_n$ in terms of these three to get these converge to some $L^2$ functions, then $(f_n,g_n,h_n)$ converges in the usual norm on that space.