Matrix Lie Groups

[hardy flame]

Might be blanking on some topology, but in the case A isnt in GL(n,C), how is G still closed in GL(n,C)? A would be a limit point outside G, so G isn't closed no?

[woeful gale]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

+close

3. Feel free to nominate the person for helper of the week in #helper-nominations

[sudden monolith]

not too familiar with this stuff, but I believe that's what the parenthetical is about

[carmine elbow]

As the text at the bottom says, G is a closed subset of GL(n,C), not of M_n(C). This means that if A_m converges in GL(n,C), then its limit point is in G. However, in the case that A isn't in G, A is non-invertible, so it's not in GL(n,C). This doesn't violate the condition of A being closed in GL(n,C), since A_m doesn't converge in GL(n,C) in that case.

[sudden monolith]

it wouldn't be closed in Mₙ(ℂ), but if the limit point isn't invertiable it isn't in GL(n, ℂ) so it may still be closed there

[sudden monolith]

ah, I missed a bit of what you said; a closed set is one that has all of its limit points, but if A isn't in GL(n, ℂ) it's not even a "point" when we talk about closed subsets of GL(n, ℂ)

[hardy flame]

oh right yeah, cause it's if every sequence in G which converges and has limit point A in GL, then A is in G

[sudden monolith]

think how (0, 1) is a closed subset of (0, 1)

[hardy flame]

yeah it's subspace top endowed onto GL