I am trying to prove $U_{p,q}$ is bounded using the induced norm $|| . ||_2$ from $M_n(\Bbb R)$ (or $M_n(\Bbb C)$ I am not sure).
A norm is an application $M_n(\Bbb C) \to \Bbb R^+$, but in the case of $U_{p,q}$ the indefinite unitary group, I made some block matrices calculations and I found that if we take $M = \begin{bmatrix} A & B \\ C & D\end{bmatrix}$ then $$ || M ||_2^2 = \mathrm{Trace}\left(\begin{bmatrix} ^t\overline{A}A+^t\overline{C}C & ^t\overline{A}B +^t\overline{C}D\\ ^t\overline{B}A+^t\overline{D}C & ^t\overline{B}B+^t\overline{D}D\end{bmatrix}\right) $$ By computing the Lie Algebra of $U_{p,q}$ we know that $A\in Lie(U_p) ,D\in Lie(U_q)$ but I don't know how to go from there to prove the trace is positive and bounded
Thank you for your help.