If $E$ is a real Hilbert space of infinite dimension, why is $GL(E)$ connected in $L(E)$?
Asked
Active
Viewed 23 times
0
-
Are you certain this is true? It false for finite-dimensional real vector spaces... – David C. Ullrich Jun 30 '19 at 14:21
-
yes it is false for finite-dimensional real vector spaces. For many Banach spaces,the group of invertible operators $GL(E)$ is connected. This is the case, for all Hilbert spaces. I need to know why it is true if $E$ is a real Hilbert spaces. – user680813 Jun 30 '19 at 14:36
-
2"This is the case, for all Hilbert spaces." ??? if you know this you're done... – David C. Ullrich Jun 30 '19 at 14:42