Completeness of eigenfunctions

Question

In my computations I have obtained a sequence of eigenvalues $\lambda_k, \; k\in \mathbb{N}$ of double multiplicity. Thus, the basis for the eigenspace of $\lambda_k$ is given by $\psi_k(x) = \left\{e^{\frac{i \pi k}{l}x}, e^{-\frac{i \pi k}{l}x}\right\}$, where $l$ is the length of the interval in which the boundary value problem was considered. My question is: what should I do in order to show that the set $\{\psi_k\}_{k\in\mathbb{N}}$ is complete in $L^2(0,l)$? Basically, I have a problem with $\psi_k$ being given by two functions, instead of one. :) Thank you in advance!