$X,Y,Z$ are Banach spaces, $A:X\to Y$ and $B:X\to Z$ are continous linear injective functions and $B$ is also compact. Moreover, there exists $C>0$ such that $\Vert x\Vert\leqslant C\Vert Ax\Vert+C\Vert Bx\Vert$. To show is there exists some constnat $D$ such that $\Vert x\Vert\leqslant D\Vert Ax\Vert$.
I would like to apply the continuous inverse theorem or open mapping theorem. But I need the image of $A$ in $Y$ is closed. However, I am not able to show it.