I'm working though Hall's Lie groups, Lie algebras, and representations and I want to show that the matrix Lie group $Sp(2N,\mathbb{R})$ is not locally compact. I've already shown that it fails to be compact since it is not bounded. The text doesn't talk about local compactness at all. How would I go about proving this?
I know that $\mathbb{R}^d$ is locally compact and subspaces need not be locally compact. To prove the space is not locally compact it would suffice to show that there exists a matrix with no compact neighborhood.
Each Lie group is a smooth manifold and each smooth manifold is locally compact.