orthogonal complement with integral as scalar product

Question

Given $V$ as the vector space of the functions that are continuous between $0$ and $2\pi$, I need to find the orthogonal complement of $$\{f\in V: f(0)=0\}$$ with the scalar product defined as: $$\langle f,g\rangle=\int_{0}^{2\pi} f(x)g(x) dx$$

Answer 1

The space $M = \{f \in C[0,2\pi] : f(0) = 0\}$ is dense in $C[0,2\pi]$. Namely, let $g \in C[0,2\pi]$ and $\varepsilon > 0$.

Set $\delta = \frac1{4\|g\|_\infty^2}$ and consider $f \in M$ which is equal to $g$ on $[\delta, 2\pi]$, and a linear function from $(0,0)$ to $(\delta, g(\delta))$ on $[0, \delta]$.

We have

$$\|f-g\|_2^2 = \int_{[0,2\pi]}|f-g|^2 = \int_{[0,\delta]} |f-g|^2 \le \int_{[0,\delta]} (2\|g\|_\infty)^2 = 4\delta \|g\|_\infty^2 < \varepsilon$$

Therefore, $\overline{M} = C[0,2\pi]$ so

$$M^\perp = \overline{M}^\perp = C[0,2\pi]^\perp = \{0\}$$