$F = \{f \in C^{1}([a,b]): f(a) = 0\}$ not complete

Question

Let $F = \{f \in C^{1}([a,b]): f(a) = 0\}$ with $\langle f, g \rangle = \int_{a}^{b} f'(x) \overline{g'(x)} dx$

Determine if $F$ with $\langle \cdot , \cdot \rangle$ is a Hilbert Space.

I believe the answer to be no. Without the $f(a) = 0$ condition, I would use something like

$f_{n}(x) = \sqrt{x + \frac{1}{n}}$ to show the space isn't complete, but this sequence wouldn't be in $F$.

Once you throw in the $f(a) = 0$ condition, I'm not sure of an appropriate Cauchy sequence. Any ideas?

Thanks.

Answer 1

Aside: The condition $f(a)=0$ is actually what tells you that $\langle f,f \rangle =0$.

For $n\in\mathbb{N}$, let $f_n \in F$ be such that, for $x>a+\frac{1}{n}$, $f_n(x)=1$ (this is possible by elementary calculus, glue a thin upside-down parabola to a constant). Then $f_n'\to 0$ pointwise, so, by the Dominated Convergence Theorem, $$ \langle f_n,f_m\rangle =\int_a^b (f_n'-f_m')^2dx\to 0 $$ as $n,m\to \infty$. But the limiting function is not continuous.