I came across a question which I think I know the answer to, but I'd like some confirmation, or a counter-example to show me where my reason fails me (as it often does....darn unreliable brain ).
the question is this:
is the subgroup of , given by:
normal in ?
the topology used on is the standard (metric) topology on , which I believe induces the frobenius norm on the matrices. this norm is sub-multiplicative, so it seems to me, that if , that:
is the desired homotopy of with , which would then prove the normality of .
the fly in the ointment being, that has to be continuous, which is where the frobenius norm comes in.
suppose that for , and denote .
if , then if I choose such that: