How to show that this vector subspace is closed?

Question

Let $A=\{f\in L^{2}[-\pi,\pi]: f\equiv 0\textrm{ in } [-\pi,0]\}$ be a vector subspace in $L^{2}[-\pi,\pi]$. I want to show that this set is closed, but I am quite lost... I have tried to express the set $A$ as $S^{\perp}$ where $S$ is a subset of $L^{2}[-\pi,\pi]$ (as I know that $S^{\perp}$ is a closed vector subspace of $L^{2}[-\pi,\pi]$). Choosing $S=\{f\chi_{[-\pi,0]}: f\in L^{2}[-\pi,\pi]\}$ was my first option but I think I am mistaken. I don't even know if this is the right approach. Any hint/help would be appreciated. Thanks in advanced!