I have encountered an interesting question, which seems to have a simple solution.
Consider $E$ as the set of $2\pi$ periodic, complex-valued, infinitely differentiable functions s.t. $\forall f\in E$, $f(0)=0$ and $$\int_{[0,2\pi]} \frac{f(x)}{x}dx = 0$$. Show that $E$ is dense in $(L^2([0, 2\pi)), \|\cdot\|_2)$ using the fact that a subspace of Hilbert space is dense iff $E^\perp = \{0\}$.
The setting of this problem reminds me of much of its Taylor Series. The fact that the function is $0$ at origin removes the first term of its expansion. I would like to use the hint to show nonzero functions $g$ would give positive inner-product with some well-chosen $f$, yet I could not do this rigorously. Could someone help?
The functions $$ e_n(x) = xe^{2i\pi nx}, \quad x \in [0,2\pi], $$ for $n \geq 1$ are in $E$. Therefore, if $f \in E^{\perp}$, then for all $n \geq 1$, $$ \int_0^{2 \pi} f(x) e_n(x) \: \mathrm{d}x = \int_0^{2 \pi} f(x) x e^{2i\pi nx} \: \mathrm{d}x = 0. $$ By this means that the function $x \mapsto x f(x)$ has all its Fourier coefficients, except the constant one, equal to $0$. This means that it is a constant function. So for some $c \in \mathbb{C}$, we have $f(x) = c/x$, but since $f \in L^2$, the only option is that $c = 0$, i.e. $f = 0$.