We denote $X_{T}$ the vector space of all $T$-periodic function with zero mean in $L^2$ ( we know that $X_{T}$ is spawn by $(e^{2i\pi nt/T})$). Let be $$X=X_{2\pi}+X_{3\pi}.$$ I think that $X_{2\pi}+X_{3\pi}$ is closed in $L^2(0,4\pi)$ but i can't prove it.