I will be honest, this question I have no clue where to even start. I know what eigenspace or gen eigen space or invariant space means but don't even know line 1 of proof. If someone can point me in the right direction or show me some ideas that would be fantastic.
To show that [imath]E_\lambda[/imath] is an invariant subspace for B one needs to show that [imath]\forall x \in E_\lambda : Bx \in E_\lambda[/imath]. From the definition of [imath]E_\lambda[/imath] it is enough to prove that if [imath]Ax=\lambda x[/imath] then [imath]ABX=\lambda BX[/imath]. But [imath]ABX=BAX
= B\lambda x = \lambda BX[/imath]. QED.