I am currently working through some lecture notes on the Geometry of Hilbert spaces, and I am stuck with the following comment:
If we are given the inner product space $C([0,1])$ of continuous functions $f : [0,1] \to \mathbb{F}$, where $\mathbb{F} = \mathbb{C}$ or $\mathbb{R}$ with the inner product \begin{equation} (f,g) = \int^1_0 f(t)\overline{g(t)} \, dt \quad f,g, \in C([0,1]) \end{equation} then the set \begin{equation}\{ e^{i2\pi nt} \}_{n \in \mathbb{Z}} \end{equation} forms a complete orthonormal set, that is, \begin{equation} (f,e^{i2\pi nt}) = 0 \quad \forall n \in \mathbb{Z} \quad \Rightarrow \quad f \equiv0 \end{equation} I tried to find an argument on my own, but somehow I get stuck each time, I try to estimate the magnitude of $f$ and derive it is zer0, but each time I get caught up in a double integral which I am sure is wrong. The proofs in notes that I found on the web all prove statements that are equivalent to the one above - I would really like to understand how to derive the completeness using only the above definition - if somebody could give me a small hint how to go about this that would be great, many thanks !