The following is a problem from a practice exam I was given for a course I am currently in:
Let $X$ be the space of all continuously differentiable functions $f : [0,1] → \Bbb{K}$ with $f(0) = 0$, endowed with the inner product $\langle f,g \rangle=\int_0^1(f(t)\overline{g(t)} + f'(t)\overline{g'(t)})dt$.
(i) Show that $\left(X,\langle \cdot, \cdot \rangle \right)$ is incomplete.
(ii) Let $H$ be the completion of X. Explain why $\phi(f):=f\left(\frac12\right)$ for all $f \in X$ can be extended to a bounded linear functional on $H$, also denoted $\phi$. Give an upper bound for $\Vert\phi\Vert$.
(iii) Determine the unique $g \in H$ such that $\phi(f) = \langle f,g \rangle$ for all $f \in H$.
I solved (i) by considering the functions $$f_n(t) := \begin{cases} 0, & t \in \left[0, \frac12-\frac1{2n}\right] \\ n(t-\frac12)+\frac12, & t \in \left[\frac12-\frac1{2n}, \frac12+\frac1{2n}\right] \\ 1, & t \in \left[\frac12+\frac1{2n},1\right] \end{cases},$$ $$g_n(t):=\int_0^tf_n(u)du$$ for $n\in \Bbb{N}$. Then $g_n(t)\to g(t):=\begin{cases} 0, & t\leq\frac12 \\ t-\frac12, & t > \frac12 \end{cases}$ as $n \to \infty$, which is not in $X$. Thus, $X$ is not closed and therefore not complete.
For (ii), by the Hahn-Banach theorem, $\phi$ can be extended to a functional on $H$ if $\phi \in X^*$, i.e. if $\Vert \phi \Vert < \infty$. However, I have not been able to find an upper bound for $\Vert \phi \Vert$.
For (iii), I am not sure how to proceed. I know, such a unique $g \in H$ exists by the Fréchet-Riesz theorem. We need to find $g \in H$ such that $$f\left(\frac12\right)=\int_0^1(f(t)\overline{g(t)} + f'(t)\overline{g'(t)})dt$$ for all $f \in H$, but I don't know where to go from here. Any help on (ii) or (iii) would be appreciated.
Of course, we only know $\phi$ can be extended after we have proven it is bounded. The upper bound on $\phi$ comes from a Poincare type inequality. Indeed, notice that \begin{align*}\phi(f) = f(1/2) &= \int^{1/2}_0 f'(x)dx \\&\le \left(\int^{1/2}_0 1^2 dx \right)^{1/2}\left(\int^{1/2}_0 \rvert f'(x)\lvert^2dx\right)^{1/2} \\&\le \frac{1}{\sqrt 2} \|f'\|_{L^2} \le \frac{1}{\sqrt 2} \|f\|_{H^1},\end{align*} where $\| \cdot \|_{H^1}$ is the norm induced by your inner product: $$\|f\|_{H^1} = \left( \int^1_0 \lvert f(x) \rvert^2 + \lvert f'(x) \rvert^2\,\, dx\right)^{1/2}.$$
For the moment, I am drawing a blank on how to find $g$ such that $\phi(f) = \langle f, g \rangle$ for all $f \in H$, but I will edit this answer if I figure it out and no other answer has been posted by then.