Sobolev space in $1-D$ is a space $H=\{f:f, f_x \in \mathcal{L}^2(-\infty, \infty)\} \subset \mathcal{L}^2(-\infty,\infty)$ with norm $||f||_{H}^2=\int_{-\infty}^{\infty}|f|^2 dx+\int_{-\infty}^{\infty}|f_x|^2 dx.$ This space is complete space w.r.t. the given norm.
Now the question is, what can we say about the completeness of the space $H$ with norm $\|f\|_{\mathcal{L}^2}^2=\int_{-\infty}^{\infty}|f|^2 dx ?$ I don't know if it will be complete or not. If it is complete, then how, and if it is not complete, then which condition should we add in $H$ to make it complete w.r.t. the norm $\|f\|_{\mathcal{L}^2}$? Boundary conditions are $\lim_{x \to \pm \infty}f(x )=0.$
I was trying to prove it by taking a Cauchy sequence say $\{f_n\}$ in $H$ which is a subset of $\mathcal{L}^2(-\infty,\infty).$ So, $\{f_n\}$ is convergent and converges to $f$ in $\mathcal{L}^2(-\infty,\infty).$ So, the remaining thing which we have to prove $f_x$ is also in $\mathcal{L}^2(-\infty,\infty)$ and derivative $\{f_n'\}$ sequence will converge to $f'$ or $f_x.$ I can prove this last step (if we assume $\{f_n'\}$ is also cauchy) but not able to prove $f_n'$ is Cauchy in $\mathcal{L}^2(-\infty,\infty).$
Thanks in advance.
You attempt cannot work. If $\mathcal{L}^2\equiv L^2$, we have that $\overline{H}^{L^2}=L^2$ as $\overline{C_c^\infty(\mathbb{R})}^{L^2}=L^2$, smooth compactly supported functions are dense in $L^2$, so $(H, \|\cdot\|_{L^2})$ is not a closed subset.
In fact, as long as $C_c^\infty(\mathbb{R})\subset H_0\subset H$, where $H_0$ is a subset of $H$ imposing certain conditions (as the ones you are asking for), we can always find a convergent subsequence $f_n\in H_0$ to $f\in L^2\backslash H$, so $H_0$ is not complete.