Let $U$ denote the set of all $n\times n$ matrices $A$ with complex entries such that $A$ is unitary. Then $U$, as a topological subspace of $\mathbb{C}^{n^2}$ is,

  1. Compact, but not connected.
  2. Connected, but not compact.
  3. Connected and compact.
  4. Neither connected nor compact.

I know what the terms are but I am really not getting the idea how to solve this. Please help.

  • 1
    $\begingroup$ For the compactness, not that we are considering a problem in some finite dimensional vector space. So compactness is equivalent to bounded and closed. Also, since every norm on this space is equivalent, choose your favourite one. $\endgroup$ – G. Chiusole Apr 2 at 8:59
  • $\begingroup$ I am sorry but I could not understand ''every norm on this space is equivalent'' and how does this specify the answer? $\endgroup$ – Huny Apr 2 at 9:00
  • $\begingroup$ Hint: consider the application $f(U):=U^{*}U$. What can you tell us about $f$ and $f^{-1}(\text{Id})$? $\endgroup$ – Caffeine Apr 2 at 9:02