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 : $$\mid g \mid_p \leq \sum_{i=1}^{\infty}\mid f_i \mid _p$$ ?
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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 $(\int |g|^{p})^{1/p} \leq \lim \inf_{N \to \infty} (\int |g_N|^{p})^{1/p} \leq \sum \|f_i\|_p$.
Kavi Rama Murthy
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Can you please remind me what is the precise theorem of Fatou ? – Dicordi Jan 16 '20 at 12:56
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Fatou's Lemma says that $\int \lim \inf f_n d\mu \leq \lim \inf \int f_n d\mu$ whenever $f_n$'s are non-negative measurable functions. @Dicordi – Kavi Rama Murthy Jan 16 '20 at 12:58
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and so here, $\lim \inf g_N=g$ true ? – Dicordi Jan 16 '20 at 13:00
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1Yes, when the limit exists, $\lim \inf$ is same as the limit. @Dicordi – Kavi Rama Murthy Jan 16 '20 at 13:01