This question may be very elementary but I want to make sure.
Many of the interpolation theorems require a linear operator to be defined on the sum of two spaces, for example $T$ is a linear operator defined on $L^p + L^r$.
However, none of the chapter exercises explicitly define the domain of the linear operator. Instead, it will read, for example, that $T$ is a linear operator that maps $L^1$ to $L^{1, \infty}$ with norm $A_0$ and $L^r$ to $L^r$ with norm $A_1$.
Can I assume then that $T$ is defined on $L^1 + L^r$?
If $T$ maps $f\in L^1$ to $Tf\in L^{1,\infty}$ and $g\in L^r$ to $Tg\in L^r$, then $T$ naturally extends to a map $$T:L^1\oplus L^r \to L^{1,\infty}\oplus L^r$$ by $$f \oplus g \mapsto Tf\oplus Tg.$$ One then verifies that this map is linear, and bounded.