Are the zeros same in $L^2$ and $\mathcal{S}'$?

Question

Let $f\in L^2(\mathbb{R}^n)$ and $g\in \mathcal{S}'(\mathbb{R}^n)$, where $\mathcal{S}'$ denotes the tempered distribution space, if $f=g$ in $\mathcal{S}'$, could we conclude $g=f$ in $L^2$?