Which of the following are compact sets?
$\{\operatorname{trace}(A): A \text{ is real orthogonal}\}$
$\{A\in M_n(\mathbb{R}):\text{ eigenvalues $|\lambda|\le 2$}\}$
Well, orthogonal matrices are compact, but the trace of them may be any $x\in\mathbb{R}$, so I guess 1 is non compact. Let $x$ be an eigenvector corresponding to the eigenvalue $\lambda$; then $Ax=\lambda x$, then $\|Ax\|= |\lambda|\cdot\|x\|\le \|A\|\cdot\|x\|$ so $\|A\|\ge 2$ so $2$ is also non compact as unbounded?