how to define bounded linear operator from $L^2+L^\infty\rightarrow L^2$

Question

how to define bounded linear operator from $L^2+L^\infty\rightarrow L^2$. Or how to define the norm in the space $L^2+L^\infty$? Has anyone any reference to this topic?

where $L^2+L^\infty=\{f|f=u+v,u\in L^2,v\in L^\infty\}$