solution of $\Delta f = 0 $ in $L^2(\mathbb{R}^n)$

Question

It is asserted that the only solution is $f = 0 $. How to prove it?

I am actually quite confused as many functions in $L^2(\mathbb{R}^n)$ are non-differentiable neither in the strong nor in the weak sense, so $\Delta f $ might not make sense for $f$.