Can an inner product space contain a complete orthonormal sequence, and still be incomplete?
This question came to my mind while reading Optimization by Vector Space Methods by David G. Luenberger. Section 3.8 of the book defines a complete orthonormal sequence. Example 2 of this section also informs that $\{e_k\} \ (k = 0, \pm 1, \pm 2, \cdots)$, where $e_k(t) = \frac{1}{\sqrt{2\pi}} e^{ikt}$ for $k = 0, \pm 1, \pm 2, \cdots$, is a complete orthonormal sequence in the complex Hilbert space $\mathscr L_2[0, 2\pi]$.
Here, I see that each $e_k$ is a complex-valued continuous mapping of $[0, 2\pi]$. Therefore $\{e_k\}$ is contained in the complex space $\mathcal C[0, 2\pi]$ consisting of all complex-valued continuous mappings of $[0, 2\pi]$. I also see that $\mathcal C[0, 2\pi]$ is a subspace in $\mathscr L_2[0, 2\pi]$, and that $\{e_k\}$ is a complete orthonormal sequence in $\mathcal C[0, 2\pi]$. (Needless to say, $\mathcal C[0, 2\pi]$ -- together with the same inner product as that of $\mathscr L_2[0, 2\pi]$ -- defines an inner product space.) However, it follows from Example 1 from section 2.11 of the book that $\mathcal C[0, 2\pi]$ is not complete. In conclusion, $\mathcal C[0, 2\pi]$ is not complete despite containing a complete orthonormal sequence.
I wonder if my conclusion is correct. If not, I would appreciate a pointer as to where my argument fails. Thanks.