Q:Suppose $L^{p_0}+L^{p_1}$ is defined as the vector space of measurable functions $f$ on a measure space $X$,that can be written as a sum $f=f_0+f_1$ with $f_0\in L^{p_0}$ and $f_1\in L^{p_1}$. Consider $$\|f\|_{L^{p_0}+L^{p_1}}=inf\{\|f_0\|_{L^{p_0}}+\|f_1\|_{L^{p_1}}\},$$where the infimum is taken over all decomposition $f=f_0+f_1$ with $f_0\in L^{p_0}$ and $f_1\in L^{p_1}$. Show that $\|\cdot\|_{L^{p_0}+L^{p_1}}$ is a norm, and that $L^{p_0}+L^{p_1}$ with this norm is a Banach space.
It is just an exercise on the functional analysis of Elias Stein. It is not difficult to prove that it is a norm. But I wonder how to deal with the completeness. If I start from the decomposition of each $f_n$ of a Cauchy sequence from the infimum. I can't get back to the norm of the difference $f_n-f_{n+p}$,and similarly if I start from the difference. I'm so confused now. May someone help me?