Compactness of unitary group in the symplectic linear group

Question

I am working on the proof of the claim, that

$$U(n) \subset Sp(2n)$$ is the maximal compact subgroup in the book "Introduction to symplectic topology" by McDuff and Salomon.

As far as I see they are only proving that $U(n)$ is maximal, is the compactness somehow trivial?

Answer 1

If you put the operator norm on the spaces of matrices, $U(n)$ is bounded (its elements have norm $1$) and it is closed, since if $A_n$ is a sequence which preserves the Hermitian metric and its limit is $A$, $A$ preserves the Hermitian metric, sot it is compact.