The question is as in the title. It has been motivated by obtaining an explicit orthonormal basis of $L^2([0,1]^3, \mathbb{R})$ as eigenvectors of $-\Delta$. Here, we assume periodic boundary conditions on $[0,1]^3$.
Let $n \in \mathbb{Z}^3$ and $x \in [0,1]^3$. Then, it is clear that the collection \begin{equation} \Bigl( \sin(2\pi n \cdot x),\cos(2\pi n \cdot x) \Bigr)_{n \in \mathbb{Z}^3} \end{equation} are eigenvectors of $-\Delta$ on $[0,1]^3$ with the eigenvalues $4\pi^2 \lvert n \rvert^2$. Moreover this collection clearly generates the Hilbert space $L^2([0,1]^3, \mathbb{R})$
However, I would like to extract some "orthonormal basis" from this generating set. For example, the sine and cosine functions corresponding to $n=(1,1,1)$ and $n=(-1,-1,-1)$ are NOT linearly independent.
At first, I thought it would be OK to just set $n \in [\mathbb{N} \cup (0)]^3$, but the situation seems much trickier...
Could anyone please help me?