Which of the following statements are true?
$(a)$ Let $n\geq 2$. The set of all matrices in $M_n(\mathbb{C})$ which represent orthogonal projections is closed in $M_n(\mathbb{C})$.
$(b)$ The set of all polynomial functions are dense in the space $L^1(]0,1[)$.
I tried to show that the preimage of the set of all orthogonal projection matrices is closed under some continuous map. But couldn't define one. For $(b)$, I think it has to do something about polynomial approximations of $L^1$ functions. But I'm not sure. So, how to proceed this one?