The question is as follows:
If $1\leq p<r\leq \infty$, prove that $L^p+L^r$ is a Banach space with norm $\lVert f\rVert= \inf\{\lVert g\rVert_p+\lVert h\rVert_r\,|\, f=g+h\in L^p+L^r\}$, and prove that for $p<q<r$, the inclusion map $L^q\to L^p+L^r$ is continuous.
So, I managed to prove $\lVert \cdot\rVert$ really is a norm, and that the result is a Banach space with the given norm, but I'm having trouble with the continuity part. Given $f\in L^q$, I considered the set $A=\{x\in X\, | \, |f(x)|>1\}$, and the functions $f\cdot 1_A\in L^p$ and $f\cdot 1_{A^c}\in L^r$ (I've already shown these inclusions). What I'm having trouble with is estimating the norms appropriately. I found that (assuming $r<\infty$)
\begin{align}
\lVert f\rVert \leq \lVert f\cdot 1_A\rVert_p + \lVert f\cdot 1_{A^c}\rVert_r \leq \lVert f\cdot 1_A \rVert_q^{q/p} + \lVert f\cdot 1_{A^c} \rVert_q^{q/r}
\end{align}
From here, I'm not sure how to get an upper bound of the form $C\lVert f\rVert_q$, for some constant $C$. Note that I've seen this answer, but I'm not sure how the last few estimates arise (particularly, why $|f\cdot 1_A|^p\leq |f\cdot 1_A|^q$ implies $\lVert f\cdot 1_A\rVert_p\leq \lVert f\cdot 1_A\rVert_q$, and likewise for the $r$ term). Any help is appreciated.
Best Answer
You are already quite far!
Recall that ~o show continuity of a linear map, you only need to show that it is continuous at $0$
If you estimate $\lVert f\cdot 1_A \rVert_q^{q/p} + \lVert f\cdot 1_{A^c} \rVert_q^{q/r}$ by $\lVert f \rVert_q^{q/p} + \lVert f\rVert_q^{q/r}$, then we have $$ \| f \|_{L^p+L^r} \to 0 \quad\text{for}\; \|f\|\to0. $$ Thus, the inclusion is continuous at $0$ and therefore continuous.