 Solution

Write the Fourier series $f(x)={\sum }_{n\in \mathbb{Z}}{f}_n{e}^{inx}$, with $f_n=(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }f(x)e^{-inx}dx$. Smoothness makes its coefficients decay faster than every inverse power. Settingtherefore gives a smooth periodic function, and termwise differentiation proves $-u_f^{\prime \prime }+u_f=f$. Conversely every solution must have these coefficients, proving uniqueness. Equivalently, a homogeneous solution has $\int (|u^{\prime }{|}^2+|u{|}^2)=0$ by integration by parts, so vanishes.