Minkowski for series

Question

Consider $g=\sum_{i=1}^{\infty}\mid f_i\mid$ where $(f_i)$ are functions on $L^p(R^n)$. I know that for every $f$ and $g$ in $L^p(R^n)$, where $p\geq 1$, we have that $\mid f+g \mid_p \leq \mid f\mid _p + \mid g\mid _p$ where $\mid . \mid_p$ is the $L^p(R^n)$ norm. Is it true that : $$\mid g \mid_p \leq \sum_{i=1}^{\infty}\mid f_i \mid _p$$ ?

Answer 1

This is immediate from Fatou's Lemma and Minkowski's inequality for finite sums. If $g_N$ is the $n-$the partial sum of $\sum |f_i|$ then $(\int |g|^{p})^{1/p} \leq \lim \inf_{N \to \infty} (\int |g_N|^{p})^{1/p} \leq \sum \|f_i\|_p$.