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**slevvio** Let $\displaystyle X$ be the space of continuous functions $\displaystyle f:[0,\pi] \rightarrow \mathbb{C}$ with norm $\displaystyle \|f\| = \int_0^\pi |f(x) |dx$. I am trying to determine whether the linearly independent set $\displaystyle \text{span}\{\sin nx\}_{n \in \mathbb{N}}$ is dense in $\displaystyle X.$ Somebody gave me a hint that it is not dense and to look at the constant function 1. So that means I am trying to show that there is some open ball around the function 1, which contains no element of $\displaystyle \text{span}\{\sin nx\}_{n \in \mathbb{N}}$.

This means that I need to show $\displaystyle \int_0^\pi |1 - \sum_{i=0}^r \alpha_i \sin (n_i x) | dx \ge \epsilon$ for some $\displaystyle \epsilon > $0.

Has anyone got any hints regarding how to go about this? Thanks very much